Wikiversity:Requests for Deletion

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        <username>Mikael Häggström</username>
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      <comment>/* Unused files uploaded by PCano */ Sounds good</comment>
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[[Category:Wikiversity deletion]]

'''&lt;big&gt;&lt;big&gt;&lt;big&gt;Deletion requests&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;'''

If an article should be deleted and does not meet [[WV:SPEEDY|speedy deletion criteria]], please list it here. Include the title and reason for deletion. If it meets speedy deletion criteria, just tag the resource with {{tlx|Delete|reason}} rather than opening a deletion discussion here.

If an article has been deleted, and you would like it undeleted, please list it here. Please try to give as close to the title as possible, and list your reasons for why it should be restored. The first line after the header should be: '''Undeletion requested'''

__TOC__

== Unused files uploaded by PCano ==
I suggest to delete the 287 unused files listed in [[:Category:Files uploaded by PCano - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2011 so it is unlikely the files will ever be used. The files seems to be a part of a set of data. I do not know if the set is complete. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:17, 26 February 2024 (UTC)
:I don't know the details, but sometimes the WikiJournals process the copyright differently. Has anybody checked with them about these files? If not, I would be happy to do the deed.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:34, 27 February 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]] I have not checked with WikiJournals. I was not thinking about copyright but if we are sure the files are correct and if they are of use to anyone? --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 27 February 2024 (UTC)
:::As I recall, files that are imbedded in pdf files are don't show up as being used.  I don't know why the WikiJournal would care,   the wikitext but want the pdf and raw files (wouldn't make any sense.) But the value of the Wikijournals is such that somebody needs to double check.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:52, 27 February 2024 (UTC)
:::: If the files are really embedded in a pdf (not linked), they are part of the pdf, and even if the files get deleted, the content is still in the pdf. What are examples of pdfs produced by Wikijurnals? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:20, 27 February 2024 (UTC)
* '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider. For the record, the files seem to be in the public domain, and many of them are for &quot;HLA allele distribution&quot;; &quot;Source: HumImmunol 2008&quot;. A selection of concerned file names: [[:File:2005 ASHI Poster 48 PCano.pdf]], [[:File:A-0101.gif]], [[:File:A-0102.gif]], [[:File:A-0103.gif]], [[:File:A-0201.gif]], [[:File:A-0202.gif]], [[:File:A-0203.gif]], [[:File:A-0204.gif]], [[:File:A-0205.gif]], [[:File:A-0206.gif]], [[:File:A-0207.gif]], [[:File:A-0208.gif]]. I randomly checked a couple of these files and they were uploaded in years 2010 and 2011. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:46, 27 February 2024 (UTC)
*Just to be safe, somebody needs to contact the WikiJournal.  This a a dormant author.  Right now my biggest problem is an active author.  I need to get an active author, [[User_talk:Saltrabook#Organizing_your_contributions|Saltrabook]], to put all their work under a single subpage before they become a bigger problem.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:29, 27 February 2024 (UTC)
*: No hurry here, AFAICT. This RFD can be opened for weeks and that is no big problem. And there are also other admins. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:50, 27 February 2024 (UTC)
*:: I made a comment at [[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]. Lets see if anyone join the discussion. And I agree that the discussion can be open for weeks. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:23, 27 February 2024 (UTC)
*:::''[[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]'' has gone unnoticed for a month. What next?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:13, 28 March 2024 (UTC)
*:::: Delete :-) In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2608383 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:18, 28 March 2024 (UTC)
: More for the record and about the question where these files were probably used: The uploader [[User:PCano]] (Pedro Cano, M.D., M.B.A. MD Anderson Cancer Center, HLA Typing Laboratory, Houston, TX ) created [[Genetics/Human Leukocyte Antigen]] (originally under the title [[HLA]], moved to [[Genetics/Human Leukocyte Antigen]] in April 2017), which was much later (in December 2022) deleted as per [[Wikiversity:Requests for Deletion/Archives/18#Subpages of Genetics/Human Leukocyte Antigen]]. Deleting the files used there seems to be a natural follow-up on that deletion decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:23, 30 March 2024 (UTC)
::{{re|Guy vandegrift}} Unless you still worry about the WikiJournals I think you can delete the files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:54, 30 April 2024 (UTC)
:::I have a meeting with the WikJournal of Science tomorrow and I will bring it up.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:57, 30 April 2024 (UTC)
::::{{ping|MGA73|Dan Polansky}} I just talked to the WikiJournal editors and they have no problem with deleting these files.  Moreover, they have no problem with deleting any unused files, with one exception: They would prefer that we not delete pdf files that are marked as preprints, without first contacting them.  These preprint pdf files are easily identified with the standard WikiJournal preprint headers.  Apparently, they keep a record of all preprints and would need to create another depository for them if the Wikiversity community decides it doesn't want to host them.  Their policy is to post the preprint pdf files only if the article is submitted for publication.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:30, 1 May 2024 (UTC)
:::::{{re|Guy vandegrift}} Thank you. I added [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] to [[:Category:WikiJournal Preprints]] to remove it from deletion suggestion [[#Unused_files_(user_uploaded_2-5_free_file_only)]]. Perhaps some one can find the right category for it? Also It could be a good idea to make sure that all the WikJournal files are categorized somewhere in [[:Category:WikiJournal]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:44, 2 May 2024 (UTC)
::::::{{re|Mikael Häggström|Evolution and evolvability|OhanaUnited}} Have I correctly conveyed the wishes of the [[WikiJournal User Group/Editors|WJ editors]] in this regard?[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:32, 2 May 2024 (UTC)
:::::::Sounds good to me, thanks! [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 12:13, 2 May 2024 (UTC)

== Archiving of Invalid fair use by User:Marshallsumter ==
* ''See [[Wikiversity:Requests for Deletion/Archives/21]]''

This space is for any unfinished business from that discussion.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 07:53, 29 February 2024 (UTC)
: Can be closed and archived, I guess. If anyone figures out a new task in the area of &quot;Invalid fair use by User:Marshallsumter&quot;, they can open a new RFD nomination as and when they do so. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:26, 30 March 2024 (UTC)
:: The problem is that the task (as mentioned in [[Wikiversity:Requests_for_Deletion/Archives/20#Pervasive copyright violations by User%3AMarshallsumter]]) is to check all the files uploaded by User:Marshallsumter and check if they meet the criteria for fair use. Sadly it is 1,151 files so I doubt anyone will spend the time on that. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:01, 30 March 2024 (UTC)
::: I tend to support preemptively deleting all files (not pages) uploaded by User:Marshallsumter. The fact that many of the files uploaded by him were determined not to meet Wikiversity criteria for fair use should be grounds enough. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:12, 30 March 2024 (UTC)
::::I thought we deleted all his files and userfied all his pages.  Apparently I was wrong: [[:File:Earth Shells to Scale.png]] // [[Earth/Geognosy/Quiz]] // [[Earth/Geognosy]]. When I deleted his images, I went to a page (category?) that someone else created. ... [https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 See also: This List].  Apparently this user spend all day long uploading files and putting them into pages he/she created. ... {{Ping|AP295}} This is why I don't bother with a couple of nutcase articles in [[Physics/Essays]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:55, 30 March 2024 (UTC)
::::: For anyone's interest, the upload list is visible at [[Special:ListFiles/Marshallsumter]]; a single-page view is at https://en.wikiversity.org/w/index.php?title=Special:ListFiles&amp;limit=1160&amp;user=Marshallsumter. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:09, 1 April 2024 (UTC)
{{outdent}} The abuse of the fair use doctrine by this former participant is so egregious that I fully support nuking all image uploads. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:25, 4 April 2024 (UTC)

:And I presume all pages by same participant that contain these images?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:09, 4 April 2024 (UTC)

::Any pages that have copious copyvio images should be deleted, along with the images. If there are pages without image violation they should be userfied. I doubt there are very many resources that have relevant learning content without copyvio. So, that leaves the resource pages open to deletion - which I support. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:44, 12 April 2024 (UTC)

=== Mixed discussion related to User:Marshallsumter and other topics ===
(Moved from [[Wikiversity:Requests_for_Deletion/Archives/22#User pages created as part of Computer Essentials (ICNS 141)]] --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC))
::{{ping|MGA73}} While I have your attention, I am confused about two lists that I compiled from various requests on RFD:
::#[https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 &gt;1500 Marshallsumter files]: ''Why we deleting Marshallsumter images?''
::#[[Draft:Original research/Literature]] &amp; [[Dominant group/Literature]] ''Marshallsumter sometimes delves into the &quot;soft&quot; (unscientific) subjects like literature where personal taste becomes important.  I see no reason to delete or even read them.''
::#{{Permalink|2608383|287 PCano files}} ''I believe these are being deleted because they are unused, yes?''
::#I am not very skilled at uploading files to commons that I did not create (most of my contributions need only attribution to other files on commons.)  I uploaded three files from the [[w:Library of Congress|loc]], and it was a time-consuming learning experience.  Is there someone else who can do it?  Perhaps I could watch till I got the hang of it.
::#After writing this I found {{Permalink|2497946#Exemption_Doctrine_Policy}}, which answers a lot of my questions.
::#I find this page a bit cluttered, but can live with it. If you want a general archiving and cleanup-just ask.
::--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:59, 2 April 2024 (UTC)
{{ping|Guy vandegrift}} Hello!
# Many of the files uploaded by Marshallsumter did not meet the requirements of fair use (violating the Exemption Doctrine Policy). I think all &quot;the easy files&quot; are deleted now. So to clean up the rest we either need hard work or a brute descision to delete everything just to be safe.
# I do not think I suggested to delete those 2 pages?
# Yes because they are unused.
# If you mean move files from here to Commons it is very easy: just click the tab &quot;Export to Wikimedia Commons&quot;. If you mean files you found on the Internet it is more tricky. You need to add the relevant information manually and more important add a source. If you found a website with hundreds or thousands of good files it may be possible to do with a bot (see [[:c:Commons:Batch uploading]]).
# Great :-)
# I can live with it too.
--[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:36, 9 April 2024 (UTC)

:On #1, I am happy with the brute decision if you are.  It's the uploader's responsibility to document the copyright.  Recently Mu301 and I &quot;rescued&quot; some high-quality photos on a high-quality resource.  But that was an exceptional case. Regarding #4, is (or should it be) our policy to move all Wikiversity files to Commons that are not fair use?  My problem with that is we sponsor some pretty low-quality stuff.  For example, instructors sometimes use Wikiversity for student submissions, and we can't delete those files until the course is over (in fact, we have no policy on deleting course-affiliated student submissions.) What do we do if the main page is a high-quality course, but some of the student submissions have no educational value?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 10 April 2024 (UTC)

::{{ping|Guy vandegrift}} I have no problem if everything is deleted in #1. And I also have no problems if course-affiliated student submissions are deleted after some time (#4). But I think both should be discussed on separate topics (perhaps just move the content to [[#Archiving_of_Invalid_fair_use_by_User:Marshallsumter]]). --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 10 April 2024 (UTC)
:::I have been on Wikiversity for more than 10 years, most of the time not paying attention to such things, but I am unaware of any policy that calls for the routine deletion of student efforts that were created as part of an established course.  If no decision has ever been made to routinely delete student efforts, we need to make sure the entire community is on board with any change in policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 10 April 2024 (UTC)
::::Yes I agree. Deleting student efforts that were created as part of an established course needs a new discussion and concensus.
::::Except if it is a copyvio then it should be deleted. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC)

=== Deleting ALL non-free uploads by User:Marshallsumter ===
Okay so it seems everyone agree that files that violates Wikiversity criteria for fair use should be deleted - not a big surprise :-D

The big question is if files should be checked one by one or if they should all be deleted. I noticed that some users more or less support to delete all non-free files.

I therefore have 2 questions:
# Do you agree to delete all non-free files?
# Would you like to try to save any of the files and if yes should all the files be put on a list or in a category or how do you propose to make that possible?

Ping [[User:Guy vandegrift]], [[User:Dan Polansky]], [[User:Mu301]], [[User:Koavf]], [[User:Omphalographer]], [[User:Dave Braunschweig]], [[User:AP295]] and [[User:MathXplore]] that was involved in discussions recently. Sorry if I missed anyone and if you do not want to join this time thats of course okay. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:51, 26 April 2024 (UTC)

:#Yes
:#No
:—[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:52, 26 April 2024 (UTC)
::Also yes to 1 and no to 2, with the understanding that this policy only applies to MS because of the large volume of images involved.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:29, 27 April 2024 (UTC)
::: Correct and also because MS had a lot of bad files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:52, 27 April 2024 (UTC)

:#yes, all non-free files should be deleted, prejudiced.
:#no, I don't believe that there is anything worth saving, in this batch from MS.
::--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 21:30, 27 April 2024 (UTC)
:: Same as Justin, Guy and mikeu: delete all Marshall Sumter-uploaded non-free files/uploads. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:20, 29 April 2024 (UTC)

== Unused files uploaded by Katluvdogs ==
{{Archive top|Consensus to delete all files. All are deleted.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:05, 12 April 2024 (UTC)}}
I suggest to delete the 137 unused files listed in [[:Category:Files uploaded by Katluvdogs - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2009 so it is unlikely the files will ever be used. The files seems to be class notes but in order for the files to be usable they have to be categorized. Also it seems many are questions and questions are good but there should also be answers somewhere in order for it to be educational. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:35, 3 March 2024 (UTC)
* (Copying from elsewhere) '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider.&lt;br/&gt;For the record, some of the files were referenced from [https://en.wikiversity.org/w/index.php?title=User:Katluvdogs/Ms.Puskarz:Class_Notes&amp;oldid=437947 this revision of User:Katluvdogs/Ms.Puskarz:Class_Notes], but the current revision of [[User:Katluvdogs/Ms.Puskarz:Class Notes]] states &quot;The website has been changed to: http://mspuskarzclassnotes.wikispaces.com/&quot;.&lt;br/&gt;On a minor note, pdfs are not a particularly good fit for a wiki, in my view.&lt;br/&gt;More for the record, a selection of the files being nominated for deletion: [[:File:3D cell model.pdf]], [[:File:Acid Rain Lab.pdf]], [[:File:Bio 16 and 17 hmwrk.pdf]], [[:File:BIO 18 H + SG.pdf]], [[:File:BIO 19 hmwrk and sg.pdf]], [[:File:BIO 20 hmwrk and sg.pdf]], [[:File:BIO 21 hmwrk.pdf]], [[:File:BIO 22 Hmwrk + sg.pdf]], [[:File:BIO 26 hmwrk + sg.pdf]], [[:File:BIO 27 hmwrk + sg.pdf]], [[:File:BIO April Calendar.pdf]], [[:File:BIO Bacteria infectious disease.pdf]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:56, 3 March 2024 (UTC)
:: In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2610073 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:20, 28 March 2024 (UTC)
:::'''Delete all files''' is my choice.  I see a 3-0 vote to delete, since Dan's vote was to delete if there are no objections (and nobody objected.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:12, 11 April 2024 (UTC)
:{{support}} deletion of all of these unused files. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:18, 12 April 2024 (UTC)
::{{done}} deleted all files in  Category:Files uploaded by Katluvdogs - unused--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:04, 12 April 2024 (UTC)
{{Archive bottom}}

==[[Ontosomose of Gender]]==
{{Archive top|Close with decision to move to [[Draft:Archive/2024/Ontosomose of Gender]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:14, 11 April 2024 (UTC)}}
I opened this RFD for a single purpose and that is: move this page created in 2007 by an anon to [[Draft:Archive/2024/Ontosomose of Gender]] rather than deleting it. It would be a pity to lose this little gem, quite possibly created as a joke. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:05, 6 March 2024 (UTC)  
:This is a good example of why we should just archive that which we do not understand.  It looks like gibberish to me, but [[w:Google Scholar|Google Scholar]] has [https://scholar.google.com/citations?user=eZgFGC0AAAAJ&amp;hl=de ''this article on him.'']  We may or may not be qualified to disagree with Google Scholar.  But we are certainly too small in number and to busy to look into everything Google Scholar deems worthy of mentioning. The suggestion that we move into [[Draft:Archive]] is seconded and {{done}}.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:06, 6 March 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]]: can you undelete [[Draft:Archive/2024/Ontosomose of Gender]], unless your intent is to actually have it deleted? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:37, 30 March 2024 (UTC)
:::I undeleted it. It was an accidental delete on my part. You may move it out of draft-archive.  Giving all editors the right to revert a move to draft-archive was my motive for creating [[Draft:Archive]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:22, 30 March 2024 (UTC)
{{Archive bottom}}

== [[OpenOffice.org]] ==

This one has me confused.  I used OpenOffice a long time ago, but grew tired of the advertising that came with the download.  The page looks good to me, but some subpages have been nominated for speedy deletion. What makes this case interesting is the [https://en.wikiversity.org/w/index.php?title=OpenOffice.org&amp;action=history ''history'']. Two high ranking WV administrators (Jtneill and Dave Braunschweig) worked hard to bring it up to speed, though I am sure neither currently objects to the project's deletion.  I drop their names so everybody believes me when I say that policy change is in the air.  Discuss it if you wish, or go ahead and make a vote so I can look for a consensus. It won't take much convincing to get me to move it to [[Draft:Archive/2024/OpenOffice.org]], especially if we leave a redirect. In fact, I will move with a redirect if anybody &quot;votes&quot; to move or delete.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:10, 7 March 2024 (UTC)
: I nominated [[OpenOffice.org/Writer]] and other subpages for speedy deletion. Looking at [[OpenOffice.org]], I do not see any saving grace either =&gt; delete, or move to userspace or move to draft archive. The page [[OpenOffice.org]] as it is does almost nothing to help one learn about OpenOffice.org; the few external links do not save it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:59, 9 March 2024 (UTC)
::I changed my vote to move relative material to WP because we don't need time-consuming solutions.  Will keep discussion open to permit others to perform the deed if they wish.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:52, 11 March 2024 (UTC)
::In the voting section I was asked why pages are safer in Draft:Archive-space than in Draft-space.  That got me thinking: Why do we have a policy that allows drafts to be deleted after 6 months?  Why not leave the effort in draft-space, with the understanding that anybody who want to improve the dormant draft can just blank it? This preserves the effort for whomever made it in the history of that draft? This will greatly reduce the number of pages that go into Draft:Archive.  I created Draft:Archive so that nobody's prior efforts would get lost.  The fewer pages I have to put there the better. We need a consensus to go into [[Wikiversity:Drafts]] and change that policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:18, 13 March 2024 (UTC)

===Voting on OpenOffice.org===
Please keep your vote, comment, and signature under 1 kB. Longer comments go in the section above.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:11, 7 March 2024 (UTC)

* '''Delete''' but move relevant material to [[w:OpenOffice]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:52, 7 March 2024 (UTC)
*'''Draft:Archive'''  (changed vote twice, now to match Dan's vote.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:20, 11 March 2024 (UTC)
*'''Draftify''' i propose this be moved to draft namespace.  or keep as is.  i see potential for this to spark creative ideas for other good faith Creative Commons content creation. Moving to draft namespace and potentially soft linking from an organizational archive page (ideally not as a sub-page) seems acceptable and sufficient. Willingness to not delete good faith contributions to the Creative Commons is greatly appreciated.  limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 17:44, 10 March 2024 (UTC) ... {{Ping|Michael Ten}} This page is safer in [[Draft:Archive/2024/OpenOffice.org]] than it is in [[Draft:OpenOffice.org]], so unless you object, I will consider your vote as a blessing to move it into Draft:Archive space.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:19, 11 March 2024 (UTC)
:: {{ping|Guy vandegrift}} I am not sure what you mean by &quot;This page is safer [...]&quot;  -- perhaps you mean it is likely likely to be effectively lost in the draft namespace or deleted from the draft namespace (?). I respect your views on that. I am happy enough that good faith contributions are moved to Draft namespace rather than deleted. I respect diversity of views and opinion about how Draft namespace could be best organized to be most collectively fruitful for the Creative Commons and this wiki. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:55, 13 March 2024 (UTC)
:::{{ping|Michael Ten}} According to [[Wikiversity:Drafts]], &quot;Resources which remain in the draft space for over 180 days (6 months) without being substantially edited may be deleted.&quot;. I do not like that policy, BTW.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:03, 13 March 2024 (UTC)
::::Interesting. Thank you for educating me on that. I agree with you; I do not think that is fruitful to the Creative Commons. You inspired [https://en.wikiversity.org/w/index.php?title=Wikiversity_talk%3ADrafts&amp;diff=2612156&amp;oldid=1998620 this suggestion]. Appreciated. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:11, 13 March 2024 (UTC)
* I wary of playing this &quot;!vote&quot; game, but I will: '''move to [[Draft:Archive]]''' or '''move to userpage''' or '''delete'''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:55, 11 March 2024 (UTC)
*I will move this to draft-archive because anybody can revert.  If nobody speaks in 10 days I will close and archive.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:22, 11 April 2024 (UTC)

==[[Metadata]]==
{{Archive top|Consens to draft-archive--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)}}
From my speedy deletion nomination: &quot;little to learn from here and the little that is here is from Wikipedia; no FR/EL&quot;. I have no objection to this being moved to user space or to [[Draft:Archive]]. Guideline: [[WV:Deletions]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:40, 9 March 2024 (UTC)
:As we decide what to do with &lt;code&gt;&lt;nowiki&gt;[[Metadata]]&lt;/nowiki&gt;&lt;/code&gt;, I assembled a choice of templates we might use in the future with such pages.  These templates use [[w:Help:Magic words|MAGIC WORDS]] that are connected to the current year and the page's location in namespace, and for that reason it is best to view the templates on a page that is actually up for deletion/pagemove. Two of the variations were designed by me to make it easier to copy/paste the new pagename (I also included the template's name to make it easier to for newbies to learn.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:59, 9 March 2024 (UTC)
:: I see only two templates as relevant: [[:Template:delete]] (speedy) and [[Template:rfd]]: non-speedy. I logged my disagreement to the template &quot;Draftify&quot; at [[Wikiversity:Colloquium#Template:Draftify]], which is what I think is the best place to discuss that template. I also created [[Wikiversity:Colloquium#Expanding WV:Deletions with Moving to Draft archive]] to codify what has recently been ongoing, namely that pages have been being moved to Draft archive instead of deleted; and I hope to get some supports there.
:: As for &quot;Metadata&quot;, the key decision is &quot;keep in mainspace&quot; vs. &quot;remove from mainspace&quot; and this is what this RFD is about.  --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:44, 10 March 2024 (UTC)
:::{{ping|Dan Polansky}} The problem with [[Wikiversity:Requests for Deletion]] is that is a [[w:Square peg in a round hole|&quot;round hole&quot; and the community is evolving towards &quot;square pegs&quot;]].  Meanwhile, I need a bottom line so I can look for sufficient consensus to act.o

===Voting on Metadata===
Change your vote as you wish. If you are not ready to vote, join the discussion directly above this &quot;voting section&quot;&quot;
*'''Delete, Draftify, or Userspace''' (&quot;vote&quot; cast on behalf of Dan Polansky by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:46, 11 March 2024 (UTC))
*'''Draftify IF the 6-month deletion rule is rescinded.'''  &lt;small&gt;Prior votes: From: Draft:Archive, to ''Delete, or Draftify'' in that order.&quot; &lt;/small&gt;--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:52, 28 March 2024 (UTC)
*''' Draftify''' (move to draft namespace) or keep as is. - Although moving to draft namespace seems acceptable and sufficient. limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:47, 13 March 2024 (UTC)
*{{done}} see [[Draft:Archive/2024/Metadata]][[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)
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==[[Facilitation]]==
Trivial questions don't save what is a page with learning outcomes that are scarce ([[WV:Deletions]]]). I don't care whether this gets deleted, moved to userspace or moved to [[Draft:Archive]]. This was proposed for deletion in 2016 by Dave Braunschweig and was &quot;saved&quot; by adding questions that in my view are trivial and do not save the article. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:01, 10 March 2024 (UTC)
: '''Delete'''. I don't think the page achieves anything. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:50, 11 March 2024 (UTC)
:'''Draftify, pending vote to rescind the 6-month draftspace deletion rule''' (latest vote change)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:25, 23 March 2024 (UTC)
:'''Archive, Delete, or Userspace''' (roughly in that order: vote cast on behalf of [[User:Dan Polansky|Dan Polansky]] by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:30, 11 March 2024 (UTC)&lt;small&gt;That's accurate. I guess I prefer Archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:57, 11 March 2024 (UTC)&lt;/small&gt;
:'''Draftify''' ('''Move to Draft namespace''') - I [https://en.wikiversity.org/w/index.php?title=Facilitation&amp;oldid=1633015 contributed to this page in good faith]. Deleting this page rather than preserving it somewhere will further decrease my motivations to contribute Creative Commons content to the Commons on this wiki, with the understanding that it is OK and considered a &quot;best practice&quot; to delete some good faith Creative Commons contributions on this wiki. A relevant rational may also be found [https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;diff=prev&amp;oldid=2611560 here]. Limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:43, 13 March 2024 (UTC)
:: The &quot;good faith&quot; talk is, in my view, entirely beside the point. Faith is not in question in deletion discussion, merely the aptness of the material for inclusion on a project, or inclusion in a specific namespace. For example, Wikiversity is not a repository of good-faith small children's creations or their analogues, or at least its mainspace is not. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:42, 16 March 2024 (UTC)
:: As an aside, the word you are looking for is &quot;rationale&quot;, not &quot;rational&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:37, 16 March 2024 (UTC)
:::{{ping|Dan Polansky}} I do not accept your premise that &quot;''Wikiversity is not a repository of (small children's creations)''&quot;. ... Also, there is a parallel discussion at [[Wikiversity_talk:Deletions#Proposed_modifications]], and it may remove most of the need for [[Draft:Archive]]. Michael Ten has pointed out that pages in draftspace could remain permanently. Looking back into the history, I discovered that I voted for the 180 limit.  I had forgotten all about that vote, but my own choice of wording jogged my memory: I voted for a 180 day limit because the decision to delete old drafts seemed like a foregone conclusion (Groupthink - who needs it!)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:51, 16 March 2024 (UTC)
:::: Well, then, from what does it follow that Wikiversity is such a repository? Which guideline, policy or scope statement? By small children I mean, say 0-6 years olds. Should e.g. scans of all pictures drawn by such children be uploadable as &quot;educational content&quot;? And if not pictures, should their first writings be uploadable? Why do they need publishing; does their local harddrive storage not serve the creative purpose enough? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:53, 16 March 2024 (UTC)
:::::I overstated my remark about children's work: For the most part, it belongs in userspace or draftspace. And, we need the parent's permission. But colleges teach courses in elementary education.  I once walked into such a course and somebody was reading a children's book to the entire class.  But we have no entrance requirements for Wikiversity, no minimum IQ is needed. Keep in mind that our differences are matters of personal taste (not factual reality.)  The question at hand at [[Wikiversity_talk:Deletions#Proposed_modifications]] is what requirements we wish to have for a page to reside indefinitely in draftspace.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:27, 16 March 2024 (UTC)
::::::I propose that we close this discussion with decision to delete, as author voted for that option.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:16, 4 April 2024 (UTC)

== [[HHF]] ==
{{Archive top|Moved to userspace--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:48, 1 April 2024 (UTC)}}
This page stands for &quot;High School Help Forum&quot;. It never became anything useful, it seems; it mostly contains pages that invite people to post but posts with actual content to learn from are missing. It has subpages that contain nothing useful, e.g. [[HHF/Physics/Introductory Physics]], [[HHF/Physics/Mechanics]], and [[HHF/Physics/Heat]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
* '''Move to Draft archive''' or '''move to user space''' or '''delete''', whatever is considered more appropriate. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
*'''Draftify (pending vote to rescind 6-month draftspace deletion rule)''' Here's my problem: (1) Moving to draft space is not possible because the effort to allow unlimited presence in draft-space is stalled. (2) I don't want to move to draft-archive space because that is more time-consuming than moving to draft space. (3) Deletion is out because I strongly oppose, and I see no evidence of a community consensus to delete (as defined by Wikipedia and Wikitionary)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:23, 23 March 2024 (UTC)
*: There can very well be a 2/3-consensus to delete if one or two people join the discussion and say something like &quot;delete per [[WV:Deletions]]&quot;. Therefore, it does not seem true that deletion is out of question. It depends on who turns out and who decides to follow the actual guideline [[WV:Deletions]] as currently specified. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:22, 23 March 2024 (UTC)
*::I took a closer look at HHF and its 39(?) subpages. It's totally empty of content, but with an interesting use of wikitext. I could transfer three or so pages to [[Draft:Archive]] and send the rest to the author's userspace.  LIke with Marshallsumpter, it would have to be moved in about three parts because I can't even move that many pages in one operation.  That makes a 2-0 vote and I'm sure nobody else would object. See also [[User_talk:MathXplore#Question_about_soft_deletion]].... --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:54, 23 March 2024 (UTC)
-----
-----
;Update&lt;br&gt;
As this RFD page has gone dormant, I will probably draft-archive this page, but leave the templates intact (i.e. I won't dewikify it.)  It has occurred to me that since pages in draft-archive are organized by year, we can slowly dewikify after a few years of no edits. I will later post details on [[Wikiversity:What-goes-where 2024]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:39, 30 March 2024 (UTC)
{{Archive bottom}}

== [[Surreal numbers]] ==
{{Archive top|Page has a makeover buy me (it was fun!). Closing as &quot;keep in mainspace&quot;-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:14, 12 April 2024 (UTC)}}
1) Initially, this page made almost no sense to me; it did not explain what the &quot;{x|y}&quot; notation was supposed to mean. 2) However, from reading [[Wikipedia: Surreal numbers]], I see this notation is actually used. But then, the Wikiversity page has very little content and does not seem to do anything that the Wikipedia page does not do better. At a minimum, it should explain the notation. The page should only exist if it does something that Wikipedia does not do, e.g. by being more didactic or tutorial-like. 3) As always, I am fine with this being moved to user space or Draft archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:05, 26 March 2024 (UTC)
:Since it has only one author, the proper place would be userspace. It could also go into subpace as a student project in mathematics. I have [[Physics/Essays]], and it could easily go there.---[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:18, 26 March 2024 (UTC)
{{Archive bottom}}

== [[Decadic numbers]] ==
Arguably, this is not good enough for the mainspace; I have no objections to this being in the draft space or the userspace. Issues: 1) The page appears to be an original research but is not marked as such; 2) it introduces the term &quot;decadic number&quot; as an original terminological invention, as far as I can tell, but does not disclose this to be the case; 3) the term &quot;decadic number&quot; is unfortunate since what is meant is something like &quot;infinite decadic number&quot;; 4) even the term &quot;number&quot; is questionable since it is not clear how these so-called numbers can have anything to do with quantity (but then, complex numbers arguably also do not express quantity, or a single quantity); 5) no attempt to formally define what a decadic number is made; this so-called decadic number appears to be a mapping from positive integers to the set of digits 0-9, to be interpreted from right to left; 6) e.g. &quot;Addition of the decadic numbers is the same as that of the integers&quot; is clearly untrue: integers are finite discrete quantities; ditto for &quot;Multiplication works the same way in the decadic numbers as in the integers&quot;.

Perhaps this can be salvaged rather than moved out of mainspace. The first thing to do is add external sources dealing with the concept or state that this is original invention; and then, address the issues. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:30, 26 March 2024 (UTC)

:As with [[Surreal numbers]] the choice is between userspace and a subspace where users could be encouraged to cooperate. Unlike Surreal numbers, I am unaware of any application in physics for this topic. The ideal place would be [[Discrete mathematics/Number theory]] because the Olympiads is a high school thing. I will contact the author about both pages--[[Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:29, 26 March 2024 (UTC)
:: If the page should stay in mainspace, I see no reason why it could not stay at [[Decadic numbers]]; I don't see moving it around in mainspace as an improvement. But my position as explained above is that it is not fit for mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:07, 26 March 2024 (UTC)
:::Decadic numbers and Surreal numbers have enough that they should be parallel subpages of the same page. I have suggested to the author that they should either create a top page, or find a top page and group these resources together.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:54, 29 March 2024 (UTC)

== [[Rational numbers/Introduction]] ==
The page does not do anything that Wikipedia does not do better: [[Wikipedia: Rational number]]. The page contains unfilled tables that seemed to be intended to explain something, but since they are empty, explain nothing. The page has no further reading, revealing no attempt to find best complementary sources online, probably of much higher quality. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:20, 26 March 2024 (UTC)
:Now I see why you were kicked off Wiktionary. Wikiversity has a long and established tradition of allowing student efforts.  This page is no worse that [[Student Projects/Major rivers in India]], a page which I randomly selected from [[Student Projects]].  I am trying to recruit students to contribute to Wikiversity.  Until the Wikiversity community changes its mind about allowing student projects, I will continue with that quest. I will change the template so as to not discourage a person clearly interested in teaching mathematics, and I want you to refrain from placing rfd templates on student efforts.  Use {{tl|subpagify}} instead.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:24, 26 March 2024 (UTC)
:: I was blocked in the English Wiktionary for &quot;racism&quot; and more. In the English Wiktionary, I often defended pages nominated for deletion  and rather rarely nominated anything for deletion. The English Wiktionary has almost no useless pages and is the 2nd most often visited project after Wikipedia. By contrast, the English Wikiversity has very few useful pages, a state of affairs that I am trying to turn around, step by step, following processes and guidelines that I did nothing to establish: [[WV:RFD]] and [[WV:Deletions]]. That is as far as persons go (ad hominem); as far as process, I hoped here to have a discussion with editors about whether this nearly useless page ([[Rational numbers/Introduction]]) should be moved out of the mainspace, and unless consensus developed for my position, I stand no chance to prevail. [[Rational numbers/Introduction]]  is not a &quot;student project&quot; in any sense of &quot;project&quot; but rather example of all-too-typical junk. Again, I do not decide, others do with me being only a single voice/input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:52, 26 March 2024 (UTC)
:::Now you are on the right track! Wikiversity might be in a transition period between allowing all sorts of pages, to morphing into a selective institution.  But the process has to change from the top-down, not from the bottom by deleting one page at a time. When I say &quot;top&quot;, I am referring not to the administrators, but to the community at large. At present, RFD has nothing near the quorum required to implement the changes you (and others) are seeking. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:05, 26 March 2024 (UTC)
:::: The only reasonable way going forward, to my mind anyway, is to follow [[WV:Deletions]] and not worry about the precedent of its countless violations. Since, should we take e.g. [[Relation between Electricity And Magnetism]], existing since 2011‎, as an example of a page to be kept, then we must keep nearly everything. There are too many pages like that, and therefore, if we take their aggregate as a binding precedent to follow, we end up in trouble, unable to delete junk. It seems only fair to proceed according [[WV:Deletions]], especially when using RFD process which gives potential opposition enough time to object. Such a procedure violates neither established guidelines nor processes; if it &quot;violates&quot; anything, then preexisting extreme lenience/tolerance toward junk, lenience that, as far as I know, was never codified into a guideline. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:29, 26 March 2024 (UTC)
:::::No. Please don't use this page as an agenda for reforming Wikiversity. Go to the Colloquium or write an essay. Having said that, I did delete [[Creating Relation between Electricity And Magnetism]] because that follows both guidelines and established practice.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:50, 26 March 2024 (UTC)
:::::Actually, lenience is given an advantage when pages are up for deletion (See [[Special:Permalink/2615245#Wikipedia's_deletion_policy]] for evidence that deletion requires somewhat of a super-majority.) But you are not calling for deletion of low quality pages. Instead you want them out of mainspace.  We have room for compromise. But, as I said before: RFD is not the place to discuss this. If you want, I could take &quot;Wikiversity:What-goes-where 2024&quot; out of my user-space and we could discuss it there.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:18, 26 March 2024 (UTC)

==[[Student Projects/Major rivers in India]]==
{{Archive top|Closing with administrative decision to keep (snowball clause) --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:14, 29 March 2024 (UTC) -premature closure reverted. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:34, 30 March 2024 (UTC)}}

This page fails [[WV:Verifiability]], for one: surely the author cannot know these statements without consulting a source, but no source (zero) is provided. Thus, the author did nothing to meet a verification standard. The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. The page uses almost no wiki features, except for boldface, so the author did not practice wiki editing either. I would have used speedy nomination, but since I expect some opposition, I go for RFD. ''This shall be my last post to this RFD nomination''; I defer to the collective of other editors for the decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
* My vote: '''Move to draft archive''' or '''Delete'''; I prefer non-deletion since then we will be able to point to this as an example of a page that has no business being in the mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
*I moved it to [[Draft:Archive/2024/Student Projects/Major rivers in India]]-and then I moved it back-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)

This topic is closed due to the [[w:simple:Wikipedia:Snowball act|''Snowball clause'']]. For more information, see {{Permalink|2617055}}-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)
: A transparent link to what above is not a Wikiversity guideline/policy: [[w:simple:Wikipedia:Snowball act]]. It says &quot;stop things which don't have a snowball's chance in hell of passing&quot;. To my mind, this is an out-of-process premature closure, but indeed, in the current Wikiversity climate, I do not seem to have &quot;a snowball's chance in hell of&quot; ensuring proper process administration. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:45, 30 March 2024 (UTC)
::The snowball clause refers to the selective deletion of on page out of 300(?) pages with the same problem.  A proposal to remove '''all''' unsourced pages in [[Student Projects]] would be a new topic and that would require a new RFD proposal, as stated in {{tl|Archive top}} 
:: Also, [[Major rivers in India]] is a subset of the bigger problem at [[Student Projects]]. It would have taken you less time to add a new topic to RFD on [[Student Projects]], than it did for me to revert my closure of this topic.  {{tl|Archive top}} instructed you to open a new project.  By inserting text into the closed topic, you obligated me to unclose it.  I think you are deliberately trying to make things difficult for me.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:08, 30 March 2024 (UTC)
::: Maybe I should have numbered the reasons for deletion. You are right that 1) a complete lack of sourcing alone would probably be not grounds enough for deletion. But there is 2) The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. Wikiversity is not a duplicate of Wikipedia (of [[W: List of major rivers of India]]); it is especially not a bad duplicate of Wikipedia. If the page was someone's half-decent attempt to write a sourced encyclopedic article, I would have probably let it be, but as it stands, this text is not worth anyone's ''reading'' time, and if it was merely an exercise in writing, it should have stayed on the local hard drive. I feel I am kind enough to this text and its author in agreeing that this can be ''moved to draft archive''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:47, 1 April 2024 (UTC)
::: Near all RFD nominations are ''selective'' in that there nearly always exist many other pages with the same or similar problem that were not yet nominated. Once multiple RFDs confirm that the problem is indeed deletion-worthy/worthy of moving out of mainspace, we may even use speedy deletion nomination, given Wikiversity's traditional RFD-phobia. (I am happy to use RFD, but I go along with WV RFD-phobia and use speedy delete as far as possible, which I feel is administratively not so nice.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:54, 1 April 2024 (UTC)
{{archive bottom}}

== [[Portal:Complex Systems Digital Campus/E-Laboratory on complex computational ecosystems/Members of the ECCE e-lab]] ==

I noticed a recent edit in the archives and stumbled upon an unanswered question by [[user:MGA73]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:29, 30 March 2024 (UTC)
: The linked page shows a list of laboratory members and their photo portraits (photos of faces). Such a thing does not seem to be particularly educational, and no big loss ensues by deletion. On the other hand, if this group of people wants to use Wikiversity to contribute research or educational material, this kind of page could be kindly tolerated. I do not really know what to do here. What is the precedent or similar previous RFD cases? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:06, 1 April 2024 (UTC)
::There is also a copyright problem and possibly a privacy issue.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:04, 1 April 2024 (UTC)
::: The page was created by [[User:Collet]] = Pierre Collet, who, believing the page, is one of three representatives of the group. Presumably, if these people did not want to be so published, they would not have agreed to Pierre's creating the page? Therefore, as for ''privacy'', should we assume a problem unless some of the members depicted contacts us, or should we rather assume Pierre Collet knew what he was doing? Pierre Collet's last edit was on 5 July 2021. Many of the images were uploaded by [[User:Pallamidessi]] in 2014, per [[Special:Contributions/Pallamidessi]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:48, 1 April 2024 (UTC)
:::: I am OK with '''keeping it as is'''.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:44, 12 April 2024 (UTC)

== [[Module:No globals]] ==
{{archive top|Deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:36, 12 April 2024 (UTC)}}
Replaced by the [[mw:Extension:Scribunto/Lua_reference_manual#strict|strict library]] of [[mw:Extension:Scribunto|Scribunto extension]]. --[[User:Liuxinyu970226|Liuxinyu970226]] ([[User talk:Liuxinyu970226|discuss]] • [[Special:Contributions/Liuxinyu970226|contribs]]) 11:40, 6 April 2024 (UTC)
:{{support}} deletion of unused and deprecated Module. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:26, 12 April 2024 (UTC)
:{{done}} Module deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:21, 12 April 2024 (UTC)
{{archive bottom}}

== Unused files (user uploaded 1 file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 1 file only)]]. There are 115 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason.

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott and Katluvdogs was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded 1 file. So it is most likely not users that were very active on Wikiversity.

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:27, 29 April 2024 (UTC)

:Many of them are not useful for any real purpose. I'll chip away at some of these. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:48, 29 April 2024 (UTC)
::It was actually pretty easy to go thru most of these as they are 1.) clearly not useful, 2.) unused, or 3.) already exported to Commons. A substantial majority has been deleted. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:11, 29 April 2024 (UTC)
:::Did them all. Mostly lo-rez selfies, images of text, and diagrams or equations that related to nothing, plus a few screenplays. None of them were in use locally, a handf
::: were already on Commons and I exported some as well.l —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:29, 29 April 2024 (UTC)

== [[Template:Cc-by-nd-3.0]] and [[:Category:CC-BY-ND-3.0]] ==
ND is not a valid license on Wikiversity and there are no pages/files using the license so I suggest to delete the template and the category. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:02, 29 April 2024 (UTC)

:{{done}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:43, 29 April 2024 (UTC)

== Unused files (user uploaded 2-5 free file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 2-5 free file only)]]. There are 81 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason. 

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott, Katluvdogs and [[#Unused files (user uploaded 1 file only)]] was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded a few free files so the files are most likely not a part of a bigger set of files. 

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future.

There are some pdf-files among. Commons does usually not value pdf-files unless they are scans of old books for example. So I do not think we should move those files unless there is a good reason to do so. 

One of the files is called &quot;[[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]]&quot; so that would fit in [[:Category:WikiJournal]]. However it is also called &quot;preprint&quot; so I'm not sure if it is the final edition. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:32, 1 May 2024 (UTC)

: I removed [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] from category and added to [[:Category:WikiJournal Preprints]] to keep it per comment [[Special:Diff/2624512]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:41, 2 May 2024 (UTC)</text>
      <sha1>4sq7y3wcmo4dxpuatnctrkxz6cwhax7</sha1>
    </revision>
    <revision>
      <id>2624766</id>
      <parentid>2624584</parentid>
      <timestamp>2024-05-02T17:16:39Z</timestamp>
      <contributor>
        <username>MGA73</username>
        <id>137394</id>
      </contributor>
      <comment>/* Unused files uploaded by PCano */ Great</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="66317" xml:space="preserve">{{/header}}
[[Category:Wikiversity deletion]]

'''&lt;big&gt;&lt;big&gt;&lt;big&gt;Deletion requests&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;'''

If an article should be deleted and does not meet [[WV:SPEEDY|speedy deletion criteria]], please list it here. Include the title and reason for deletion. If it meets speedy deletion criteria, just tag the resource with {{tlx|Delete|reason}} rather than opening a deletion discussion here.

If an article has been deleted, and you would like it undeleted, please list it here. Please try to give as close to the title as possible, and list your reasons for why it should be restored. The first line after the header should be: '''Undeletion requested'''

__TOC__

== Unused files uploaded by PCano ==
I suggest to delete the 287 unused files listed in [[:Category:Files uploaded by PCano - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2011 so it is unlikely the files will ever be used. The files seems to be a part of a set of data. I do not know if the set is complete. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:17, 26 February 2024 (UTC)
:I don't know the details, but sometimes the WikiJournals process the copyright differently. Has anybody checked with them about these files? If not, I would be happy to do the deed.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:34, 27 February 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]] I have not checked with WikiJournals. I was not thinking about copyright but if we are sure the files are correct and if they are of use to anyone? --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 27 February 2024 (UTC)
:::As I recall, files that are imbedded in pdf files are don't show up as being used.  I don't know why the WikiJournal would care,   the wikitext but want the pdf and raw files (wouldn't make any sense.) But the value of the Wikijournals is such that somebody needs to double check.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:52, 27 February 2024 (UTC)
:::: If the files are really embedded in a pdf (not linked), they are part of the pdf, and even if the files get deleted, the content is still in the pdf. What are examples of pdfs produced by Wikijurnals? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:20, 27 February 2024 (UTC)
* '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider. For the record, the files seem to be in the public domain, and many of them are for &quot;HLA allele distribution&quot;; &quot;Source: HumImmunol 2008&quot;. A selection of concerned file names: [[:File:2005 ASHI Poster 48 PCano.pdf]], [[:File:A-0101.gif]], [[:File:A-0102.gif]], [[:File:A-0103.gif]], [[:File:A-0201.gif]], [[:File:A-0202.gif]], [[:File:A-0203.gif]], [[:File:A-0204.gif]], [[:File:A-0205.gif]], [[:File:A-0206.gif]], [[:File:A-0207.gif]], [[:File:A-0208.gif]]. I randomly checked a couple of these files and they were uploaded in years 2010 and 2011. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:46, 27 February 2024 (UTC)
*Just to be safe, somebody needs to contact the WikiJournal.  This a a dormant author.  Right now my biggest problem is an active author.  I need to get an active author, [[User_talk:Saltrabook#Organizing_your_contributions|Saltrabook]], to put all their work under a single subpage before they become a bigger problem.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:29, 27 February 2024 (UTC)
*: No hurry here, AFAICT. This RFD can be opened for weeks and that is no big problem. And there are also other admins. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:50, 27 February 2024 (UTC)
*:: I made a comment at [[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]. Lets see if anyone join the discussion. And I agree that the discussion can be open for weeks. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:23, 27 February 2024 (UTC)
*:::''[[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]'' has gone unnoticed for a month. What next?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:13, 28 March 2024 (UTC)
*:::: Delete :-) In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2608383 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:18, 28 March 2024 (UTC)
: More for the record and about the question where these files were probably used: The uploader [[User:PCano]] (Pedro Cano, M.D., M.B.A. MD Anderson Cancer Center, HLA Typing Laboratory, Houston, TX ) created [[Genetics/Human Leukocyte Antigen]] (originally under the title [[HLA]], moved to [[Genetics/Human Leukocyte Antigen]] in April 2017), which was much later (in December 2022) deleted as per [[Wikiversity:Requests for Deletion/Archives/18#Subpages of Genetics/Human Leukocyte Antigen]]. Deleting the files used there seems to be a natural follow-up on that deletion decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:23, 30 March 2024 (UTC)
::{{re|Guy vandegrift}} Unless you still worry about the WikiJournals I think you can delete the files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:54, 30 April 2024 (UTC)
:::I have a meeting with the WikJournal of Science tomorrow and I will bring it up.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:57, 30 April 2024 (UTC)
::::{{ping|MGA73|Dan Polansky}} I just talked to the WikiJournal editors and they have no problem with deleting these files.  Moreover, they have no problem with deleting any unused files, with one exception: They would prefer that we not delete pdf files that are marked as preprints, without first contacting them.  These preprint pdf files are easily identified with the standard WikiJournal preprint headers.  Apparently, they keep a record of all preprints and would need to create another depository for them if the Wikiversity community decides it doesn't want to host them.  Their policy is to post the preprint pdf files only if the article is submitted for publication.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:30, 1 May 2024 (UTC)
:::::{{re|Guy vandegrift}} Thank you. I added [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] to [[:Category:WikiJournal Preprints]] to remove it from deletion suggestion [[#Unused_files_(user_uploaded_2-5_free_file_only)]]. Perhaps some one can find the right category for it? Also It could be a good idea to make sure that all the WikJournal files are categorized somewhere in [[:Category:WikiJournal]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:44, 2 May 2024 (UTC)
::::::{{re|Mikael Häggström|Evolution and evolvability|OhanaUnited}} Have I correctly conveyed the wishes of the [[WikiJournal User Group/Editors|WJ editors]] in this regard?[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:32, 2 May 2024 (UTC)
:::::::Sounds good to me, thanks! [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 12:13, 2 May 2024 (UTC)
:::::::: Great! And as info I can tell that I made by bot add all files that seems to be related in any way to [[:Category:WikiJournal]]. For example if the word WikiJournal is used on the file page or the file is used on a page with WikiJournal in the title. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:16, 2 May 2024 (UTC)

== Archiving of Invalid fair use by User:Marshallsumter ==
* ''See [[Wikiversity:Requests for Deletion/Archives/21]]''

This space is for any unfinished business from that discussion.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 07:53, 29 February 2024 (UTC)
: Can be closed and archived, I guess. If anyone figures out a new task in the area of &quot;Invalid fair use by User:Marshallsumter&quot;, they can open a new RFD nomination as and when they do so. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:26, 30 March 2024 (UTC)
:: The problem is that the task (as mentioned in [[Wikiversity:Requests_for_Deletion/Archives/20#Pervasive copyright violations by User%3AMarshallsumter]]) is to check all the files uploaded by User:Marshallsumter and check if they meet the criteria for fair use. Sadly it is 1,151 files so I doubt anyone will spend the time on that. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:01, 30 March 2024 (UTC)
::: I tend to support preemptively deleting all files (not pages) uploaded by User:Marshallsumter. The fact that many of the files uploaded by him were determined not to meet Wikiversity criteria for fair use should be grounds enough. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:12, 30 March 2024 (UTC)
::::I thought we deleted all his files and userfied all his pages.  Apparently I was wrong: [[:File:Earth Shells to Scale.png]] // [[Earth/Geognosy/Quiz]] // [[Earth/Geognosy]]. When I deleted his images, I went to a page (category?) that someone else created. ... [https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 See also: This List].  Apparently this user spend all day long uploading files and putting them into pages he/she created. ... {{Ping|AP295}} This is why I don't bother with a couple of nutcase articles in [[Physics/Essays]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:55, 30 March 2024 (UTC)
::::: For anyone's interest, the upload list is visible at [[Special:ListFiles/Marshallsumter]]; a single-page view is at https://en.wikiversity.org/w/index.php?title=Special:ListFiles&amp;limit=1160&amp;user=Marshallsumter. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:09, 1 April 2024 (UTC)
{{outdent}} The abuse of the fair use doctrine by this former participant is so egregious that I fully support nuking all image uploads. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:25, 4 April 2024 (UTC)

:And I presume all pages by same participant that contain these images?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:09, 4 April 2024 (UTC)

::Any pages that have copious copyvio images should be deleted, along with the images. If there are pages without image violation they should be userfied. I doubt there are very many resources that have relevant learning content without copyvio. So, that leaves the resource pages open to deletion - which I support. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:44, 12 April 2024 (UTC)

=== Mixed discussion related to User:Marshallsumter and other topics ===
(Moved from [[Wikiversity:Requests_for_Deletion/Archives/22#User pages created as part of Computer Essentials (ICNS 141)]] --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC))
::{{ping|MGA73}} While I have your attention, I am confused about two lists that I compiled from various requests on RFD:
::#[https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 &gt;1500 Marshallsumter files]: ''Why we deleting Marshallsumter images?''
::#[[Draft:Original research/Literature]] &amp; [[Dominant group/Literature]] ''Marshallsumter sometimes delves into the &quot;soft&quot; (unscientific) subjects like literature where personal taste becomes important.  I see no reason to delete or even read them.''
::#{{Permalink|2608383|287 PCano files}} ''I believe these are being deleted because they are unused, yes?''
::#I am not very skilled at uploading files to commons that I did not create (most of my contributions need only attribution to other files on commons.)  I uploaded three files from the [[w:Library of Congress|loc]], and it was a time-consuming learning experience.  Is there someone else who can do it?  Perhaps I could watch till I got the hang of it.
::#After writing this I found {{Permalink|2497946#Exemption_Doctrine_Policy}}, which answers a lot of my questions.
::#I find this page a bit cluttered, but can live with it. If you want a general archiving and cleanup-just ask.
::--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:59, 2 April 2024 (UTC)
{{ping|Guy vandegrift}} Hello!
# Many of the files uploaded by Marshallsumter did not meet the requirements of fair use (violating the Exemption Doctrine Policy). I think all &quot;the easy files&quot; are deleted now. So to clean up the rest we either need hard work or a brute descision to delete everything just to be safe.
# I do not think I suggested to delete those 2 pages?
# Yes because they are unused.
# If you mean move files from here to Commons it is very easy: just click the tab &quot;Export to Wikimedia Commons&quot;. If you mean files you found on the Internet it is more tricky. You need to add the relevant information manually and more important add a source. If you found a website with hundreds or thousands of good files it may be possible to do with a bot (see [[:c:Commons:Batch uploading]]).
# Great :-)
# I can live with it too.
--[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:36, 9 April 2024 (UTC)

:On #1, I am happy with the brute decision if you are.  It's the uploader's responsibility to document the copyright.  Recently Mu301 and I &quot;rescued&quot; some high-quality photos on a high-quality resource.  But that was an exceptional case. Regarding #4, is (or should it be) our policy to move all Wikiversity files to Commons that are not fair use?  My problem with that is we sponsor some pretty low-quality stuff.  For example, instructors sometimes use Wikiversity for student submissions, and we can't delete those files until the course is over (in fact, we have no policy on deleting course-affiliated student submissions.) What do we do if the main page is a high-quality course, but some of the student submissions have no educational value?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 10 April 2024 (UTC)

::{{ping|Guy vandegrift}} I have no problem if everything is deleted in #1. And I also have no problems if course-affiliated student submissions are deleted after some time (#4). But I think both should be discussed on separate topics (perhaps just move the content to [[#Archiving_of_Invalid_fair_use_by_User:Marshallsumter]]). --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 10 April 2024 (UTC)
:::I have been on Wikiversity for more than 10 years, most of the time not paying attention to such things, but I am unaware of any policy that calls for the routine deletion of student efforts that were created as part of an established course.  If no decision has ever been made to routinely delete student efforts, we need to make sure the entire community is on board with any change in policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 10 April 2024 (UTC)
::::Yes I agree. Deleting student efforts that were created as part of an established course needs a new discussion and concensus.
::::Except if it is a copyvio then it should be deleted. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC)

=== Deleting ALL non-free uploads by User:Marshallsumter ===
Okay so it seems everyone agree that files that violates Wikiversity criteria for fair use should be deleted - not a big surprise :-D

The big question is if files should be checked one by one or if they should all be deleted. I noticed that some users more or less support to delete all non-free files.

I therefore have 2 questions:
# Do you agree to delete all non-free files?
# Would you like to try to save any of the files and if yes should all the files be put on a list or in a category or how do you propose to make that possible?

Ping [[User:Guy vandegrift]], [[User:Dan Polansky]], [[User:Mu301]], [[User:Koavf]], [[User:Omphalographer]], [[User:Dave Braunschweig]], [[User:AP295]] and [[User:MathXplore]] that was involved in discussions recently. Sorry if I missed anyone and if you do not want to join this time thats of course okay. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:51, 26 April 2024 (UTC)

:#Yes
:#No
:—[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:52, 26 April 2024 (UTC)
::Also yes to 1 and no to 2, with the understanding that this policy only applies to MS because of the large volume of images involved.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:29, 27 April 2024 (UTC)
::: Correct and also because MS had a lot of bad files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:52, 27 April 2024 (UTC)

:#yes, all non-free files should be deleted, prejudiced.
:#no, I don't believe that there is anything worth saving, in this batch from MS.
::--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 21:30, 27 April 2024 (UTC)
:: Same as Justin, Guy and mikeu: delete all Marshall Sumter-uploaded non-free files/uploads. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:20, 29 April 2024 (UTC)

== Unused files uploaded by Katluvdogs ==
{{Archive top|Consensus to delete all files. All are deleted.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:05, 12 April 2024 (UTC)}}
I suggest to delete the 137 unused files listed in [[:Category:Files uploaded by Katluvdogs - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2009 so it is unlikely the files will ever be used. The files seems to be class notes but in order for the files to be usable they have to be categorized. Also it seems many are questions and questions are good but there should also be answers somewhere in order for it to be educational. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:35, 3 March 2024 (UTC)
* (Copying from elsewhere) '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider.&lt;br/&gt;For the record, some of the files were referenced from [https://en.wikiversity.org/w/index.php?title=User:Katluvdogs/Ms.Puskarz:Class_Notes&amp;oldid=437947 this revision of User:Katluvdogs/Ms.Puskarz:Class_Notes], but the current revision of [[User:Katluvdogs/Ms.Puskarz:Class Notes]] states &quot;The website has been changed to: http://mspuskarzclassnotes.wikispaces.com/&quot;.&lt;br/&gt;On a minor note, pdfs are not a particularly good fit for a wiki, in my view.&lt;br/&gt;More for the record, a selection of the files being nominated for deletion: [[:File:3D cell model.pdf]], [[:File:Acid Rain Lab.pdf]], [[:File:Bio 16 and 17 hmwrk.pdf]], [[:File:BIO 18 H + SG.pdf]], [[:File:BIO 19 hmwrk and sg.pdf]], [[:File:BIO 20 hmwrk and sg.pdf]], [[:File:BIO 21 hmwrk.pdf]], [[:File:BIO 22 Hmwrk + sg.pdf]], [[:File:BIO 26 hmwrk + sg.pdf]], [[:File:BIO 27 hmwrk + sg.pdf]], [[:File:BIO April Calendar.pdf]], [[:File:BIO Bacteria infectious disease.pdf]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:56, 3 March 2024 (UTC)
:: In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2610073 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:20, 28 March 2024 (UTC)
:::'''Delete all files''' is my choice.  I see a 3-0 vote to delete, since Dan's vote was to delete if there are no objections (and nobody objected.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:12, 11 April 2024 (UTC)
:{{support}} deletion of all of these unused files. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:18, 12 April 2024 (UTC)
::{{done}} deleted all files in  Category:Files uploaded by Katluvdogs - unused--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:04, 12 April 2024 (UTC)
{{Archive bottom}}

==[[Ontosomose of Gender]]==
{{Archive top|Close with decision to move to [[Draft:Archive/2024/Ontosomose of Gender]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:14, 11 April 2024 (UTC)}}
I opened this RFD for a single purpose and that is: move this page created in 2007 by an anon to [[Draft:Archive/2024/Ontosomose of Gender]] rather than deleting it. It would be a pity to lose this little gem, quite possibly created as a joke. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:05, 6 March 2024 (UTC)  
:This is a good example of why we should just archive that which we do not understand.  It looks like gibberish to me, but [[w:Google Scholar|Google Scholar]] has [https://scholar.google.com/citations?user=eZgFGC0AAAAJ&amp;hl=de ''this article on him.'']  We may or may not be qualified to disagree with Google Scholar.  But we are certainly too small in number and to busy to look into everything Google Scholar deems worthy of mentioning. The suggestion that we move into [[Draft:Archive]] is seconded and {{done}}.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:06, 6 March 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]]: can you undelete [[Draft:Archive/2024/Ontosomose of Gender]], unless your intent is to actually have it deleted? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:37, 30 March 2024 (UTC)
:::I undeleted it. It was an accidental delete on my part. You may move it out of draft-archive.  Giving all editors the right to revert a move to draft-archive was my motive for creating [[Draft:Archive]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:22, 30 March 2024 (UTC)
{{Archive bottom}}

== [[OpenOffice.org]] ==

This one has me confused.  I used OpenOffice a long time ago, but grew tired of the advertising that came with the download.  The page looks good to me, but some subpages have been nominated for speedy deletion. What makes this case interesting is the [https://en.wikiversity.org/w/index.php?title=OpenOffice.org&amp;action=history ''history'']. Two high ranking WV administrators (Jtneill and Dave Braunschweig) worked hard to bring it up to speed, though I am sure neither currently objects to the project's deletion.  I drop their names so everybody believes me when I say that policy change is in the air.  Discuss it if you wish, or go ahead and make a vote so I can look for a consensus. It won't take much convincing to get me to move it to [[Draft:Archive/2024/OpenOffice.org]], especially if we leave a redirect. In fact, I will move with a redirect if anybody &quot;votes&quot; to move or delete.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:10, 7 March 2024 (UTC)
: I nominated [[OpenOffice.org/Writer]] and other subpages for speedy deletion. Looking at [[OpenOffice.org]], I do not see any saving grace either =&gt; delete, or move to userspace or move to draft archive. The page [[OpenOffice.org]] as it is does almost nothing to help one learn about OpenOffice.org; the few external links do not save it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:59, 9 March 2024 (UTC)
::I changed my vote to move relative material to WP because we don't need time-consuming solutions.  Will keep discussion open to permit others to perform the deed if they wish.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:52, 11 March 2024 (UTC)
::In the voting section I was asked why pages are safer in Draft:Archive-space than in Draft-space.  That got me thinking: Why do we have a policy that allows drafts to be deleted after 6 months?  Why not leave the effort in draft-space, with the understanding that anybody who want to improve the dormant draft can just blank it? This preserves the effort for whomever made it in the history of that draft? This will greatly reduce the number of pages that go into Draft:Archive.  I created Draft:Archive so that nobody's prior efforts would get lost.  The fewer pages I have to put there the better. We need a consensus to go into [[Wikiversity:Drafts]] and change that policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:18, 13 March 2024 (UTC)

===Voting on OpenOffice.org===
Please keep your vote, comment, and signature under 1 kB. Longer comments go in the section above.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:11, 7 March 2024 (UTC)

* '''Delete''' but move relevant material to [[w:OpenOffice]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:52, 7 March 2024 (UTC)
*'''Draft:Archive'''  (changed vote twice, now to match Dan's vote.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:20, 11 March 2024 (UTC)
*'''Draftify''' i propose this be moved to draft namespace.  or keep as is.  i see potential for this to spark creative ideas for other good faith Creative Commons content creation. Moving to draft namespace and potentially soft linking from an organizational archive page (ideally not as a sub-page) seems acceptable and sufficient. Willingness to not delete good faith contributions to the Creative Commons is greatly appreciated.  limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 17:44, 10 March 2024 (UTC) ... {{Ping|Michael Ten}} This page is safer in [[Draft:Archive/2024/OpenOffice.org]] than it is in [[Draft:OpenOffice.org]], so unless you object, I will consider your vote as a blessing to move it into Draft:Archive space.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:19, 11 March 2024 (UTC)
:: {{ping|Guy vandegrift}} I am not sure what you mean by &quot;This page is safer [...]&quot;  -- perhaps you mean it is likely likely to be effectively lost in the draft namespace or deleted from the draft namespace (?). I respect your views on that. I am happy enough that good faith contributions are moved to Draft namespace rather than deleted. I respect diversity of views and opinion about how Draft namespace could be best organized to be most collectively fruitful for the Creative Commons and this wiki. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:55, 13 March 2024 (UTC)
:::{{ping|Michael Ten}} According to [[Wikiversity:Drafts]], &quot;Resources which remain in the draft space for over 180 days (6 months) without being substantially edited may be deleted.&quot;. I do not like that policy, BTW.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:03, 13 March 2024 (UTC)
::::Interesting. Thank you for educating me on that. I agree with you; I do not think that is fruitful to the Creative Commons. You inspired [https://en.wikiversity.org/w/index.php?title=Wikiversity_talk%3ADrafts&amp;diff=2612156&amp;oldid=1998620 this suggestion]. Appreciated. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:11, 13 March 2024 (UTC)
* I wary of playing this &quot;!vote&quot; game, but I will: '''move to [[Draft:Archive]]''' or '''move to userpage''' or '''delete'''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:55, 11 March 2024 (UTC)
*I will move this to draft-archive because anybody can revert.  If nobody speaks in 10 days I will close and archive.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:22, 11 April 2024 (UTC)

==[[Metadata]]==
{{Archive top|Consens to draft-archive--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)}}
From my speedy deletion nomination: &quot;little to learn from here and the little that is here is from Wikipedia; no FR/EL&quot;. I have no objection to this being moved to user space or to [[Draft:Archive]]. Guideline: [[WV:Deletions]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:40, 9 March 2024 (UTC)
:As we decide what to do with &lt;code&gt;&lt;nowiki&gt;[[Metadata]]&lt;/nowiki&gt;&lt;/code&gt;, I assembled a choice of templates we might use in the future with such pages.  These templates use [[w:Help:Magic words|MAGIC WORDS]] that are connected to the current year and the page's location in namespace, and for that reason it is best to view the templates on a page that is actually up for deletion/pagemove. Two of the variations were designed by me to make it easier to copy/paste the new pagename (I also included the template's name to make it easier to for newbies to learn.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:59, 9 March 2024 (UTC)
:: I see only two templates as relevant: [[:Template:delete]] (speedy) and [[Template:rfd]]: non-speedy. I logged my disagreement to the template &quot;Draftify&quot; at [[Wikiversity:Colloquium#Template:Draftify]], which is what I think is the best place to discuss that template. I also created [[Wikiversity:Colloquium#Expanding WV:Deletions with Moving to Draft archive]] to codify what has recently been ongoing, namely that pages have been being moved to Draft archive instead of deleted; and I hope to get some supports there.
:: As for &quot;Metadata&quot;, the key decision is &quot;keep in mainspace&quot; vs. &quot;remove from mainspace&quot; and this is what this RFD is about.  --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:44, 10 March 2024 (UTC)
:::{{ping|Dan Polansky}} The problem with [[Wikiversity:Requests for Deletion]] is that is a [[w:Square peg in a round hole|&quot;round hole&quot; and the community is evolving towards &quot;square pegs&quot;]].  Meanwhile, I need a bottom line so I can look for sufficient consensus to act.o

===Voting on Metadata===
Change your vote as you wish. If you are not ready to vote, join the discussion directly above this &quot;voting section&quot;&quot;
*'''Delete, Draftify, or Userspace''' (&quot;vote&quot; cast on behalf of Dan Polansky by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:46, 11 March 2024 (UTC))
*'''Draftify IF the 6-month deletion rule is rescinded.'''  &lt;small&gt;Prior votes: From: Draft:Archive, to ''Delete, or Draftify'' in that order.&quot; &lt;/small&gt;--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:52, 28 March 2024 (UTC)
*''' Draftify''' (move to draft namespace) or keep as is. - Although moving to draft namespace seems acceptable and sufficient. limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:47, 13 March 2024 (UTC)
*{{done}} see [[Draft:Archive/2024/Metadata]][[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)
{{Archive bottom}}

==[[Facilitation]]==
Trivial questions don't save what is a page with learning outcomes that are scarce ([[WV:Deletions]]]). I don't care whether this gets deleted, moved to userspace or moved to [[Draft:Archive]]. This was proposed for deletion in 2016 by Dave Braunschweig and was &quot;saved&quot; by adding questions that in my view are trivial and do not save the article. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:01, 10 March 2024 (UTC)
: '''Delete'''. I don't think the page achieves anything. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:50, 11 March 2024 (UTC)
:'''Draftify, pending vote to rescind the 6-month draftspace deletion rule''' (latest vote change)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:25, 23 March 2024 (UTC)
:'''Archive, Delete, or Userspace''' (roughly in that order: vote cast on behalf of [[User:Dan Polansky|Dan Polansky]] by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:30, 11 March 2024 (UTC)&lt;small&gt;That's accurate. I guess I prefer Archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:57, 11 March 2024 (UTC)&lt;/small&gt;
:'''Draftify''' ('''Move to Draft namespace''') - I [https://en.wikiversity.org/w/index.php?title=Facilitation&amp;oldid=1633015 contributed to this page in good faith]. Deleting this page rather than preserving it somewhere will further decrease my motivations to contribute Creative Commons content to the Commons on this wiki, with the understanding that it is OK and considered a &quot;best practice&quot; to delete some good faith Creative Commons contributions on this wiki. A relevant rational may also be found [https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;diff=prev&amp;oldid=2611560 here]. Limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:43, 13 March 2024 (UTC)
:: The &quot;good faith&quot; talk is, in my view, entirely beside the point. Faith is not in question in deletion discussion, merely the aptness of the material for inclusion on a project, or inclusion in a specific namespace. For example, Wikiversity is not a repository of good-faith small children's creations or their analogues, or at least its mainspace is not. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:42, 16 March 2024 (UTC)
:: As an aside, the word you are looking for is &quot;rationale&quot;, not &quot;rational&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:37, 16 March 2024 (UTC)
:::{{ping|Dan Polansky}} I do not accept your premise that &quot;''Wikiversity is not a repository of (small children's creations)''&quot;. ... Also, there is a parallel discussion at [[Wikiversity_talk:Deletions#Proposed_modifications]], and it may remove most of the need for [[Draft:Archive]]. Michael Ten has pointed out that pages in draftspace could remain permanently. Looking back into the history, I discovered that I voted for the 180 limit.  I had forgotten all about that vote, but my own choice of wording jogged my memory: I voted for a 180 day limit because the decision to delete old drafts seemed like a foregone conclusion (Groupthink - who needs it!)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:51, 16 March 2024 (UTC)
:::: Well, then, from what does it follow that Wikiversity is such a repository? Which guideline, policy or scope statement? By small children I mean, say 0-6 years olds. Should e.g. scans of all pictures drawn by such children be uploadable as &quot;educational content&quot;? And if not pictures, should their first writings be uploadable? Why do they need publishing; does their local harddrive storage not serve the creative purpose enough? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:53, 16 March 2024 (UTC)
:::::I overstated my remark about children's work: For the most part, it belongs in userspace or draftspace. And, we need the parent's permission. But colleges teach courses in elementary education.  I once walked into such a course and somebody was reading a children's book to the entire class.  But we have no entrance requirements for Wikiversity, no minimum IQ is needed. Keep in mind that our differences are matters of personal taste (not factual reality.)  The question at hand at [[Wikiversity_talk:Deletions#Proposed_modifications]] is what requirements we wish to have for a page to reside indefinitely in draftspace.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:27, 16 March 2024 (UTC)
::::::I propose that we close this discussion with decision to delete, as author voted for that option.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:16, 4 April 2024 (UTC)

== [[HHF]] ==
{{Archive top|Moved to userspace--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:48, 1 April 2024 (UTC)}}
This page stands for &quot;High School Help Forum&quot;. It never became anything useful, it seems; it mostly contains pages that invite people to post but posts with actual content to learn from are missing. It has subpages that contain nothing useful, e.g. [[HHF/Physics/Introductory Physics]], [[HHF/Physics/Mechanics]], and [[HHF/Physics/Heat]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
* '''Move to Draft archive''' or '''move to user space''' or '''delete''', whatever is considered more appropriate. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
*'''Draftify (pending vote to rescind 6-month draftspace deletion rule)''' Here's my problem: (1) Moving to draft space is not possible because the effort to allow unlimited presence in draft-space is stalled. (2) I don't want to move to draft-archive space because that is more time-consuming than moving to draft space. (3) Deletion is out because I strongly oppose, and I see no evidence of a community consensus to delete (as defined by Wikipedia and Wikitionary)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:23, 23 March 2024 (UTC)
*: There can very well be a 2/3-consensus to delete if one or two people join the discussion and say something like &quot;delete per [[WV:Deletions]]&quot;. Therefore, it does not seem true that deletion is out of question. It depends on who turns out and who decides to follow the actual guideline [[WV:Deletions]] as currently specified. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:22, 23 March 2024 (UTC)
*::I took a closer look at HHF and its 39(?) subpages. It's totally empty of content, but with an interesting use of wikitext. I could transfer three or so pages to [[Draft:Archive]] and send the rest to the author's userspace.  LIke with Marshallsumpter, it would have to be moved in about three parts because I can't even move that many pages in one operation.  That makes a 2-0 vote and I'm sure nobody else would object. See also [[User_talk:MathXplore#Question_about_soft_deletion]].... --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:54, 23 March 2024 (UTC)
-----
-----
;Update&lt;br&gt;
As this RFD page has gone dormant, I will probably draft-archive this page, but leave the templates intact (i.e. I won't dewikify it.)  It has occurred to me that since pages in draft-archive are organized by year, we can slowly dewikify after a few years of no edits. I will later post details on [[Wikiversity:What-goes-where 2024]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:39, 30 March 2024 (UTC)
{{Archive bottom}}

== [[Surreal numbers]] ==
{{Archive top|Page has a makeover buy me (it was fun!). Closing as &quot;keep in mainspace&quot;-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:14, 12 April 2024 (UTC)}}
1) Initially, this page made almost no sense to me; it did not explain what the &quot;{x|y}&quot; notation was supposed to mean. 2) However, from reading [[Wikipedia: Surreal numbers]], I see this notation is actually used. But then, the Wikiversity page has very little content and does not seem to do anything that the Wikipedia page does not do better. At a minimum, it should explain the notation. The page should only exist if it does something that Wikipedia does not do, e.g. by being more didactic or tutorial-like. 3) As always, I am fine with this being moved to user space or Draft archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:05, 26 March 2024 (UTC)
:Since it has only one author, the proper place would be userspace. It could also go into subpace as a student project in mathematics. I have [[Physics/Essays]], and it could easily go there.---[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:18, 26 March 2024 (UTC)
{{Archive bottom}}

== [[Decadic numbers]] ==
Arguably, this is not good enough for the mainspace; I have no objections to this being in the draft space or the userspace. Issues: 1) The page appears to be an original research but is not marked as such; 2) it introduces the term &quot;decadic number&quot; as an original terminological invention, as far as I can tell, but does not disclose this to be the case; 3) the term &quot;decadic number&quot; is unfortunate since what is meant is something like &quot;infinite decadic number&quot;; 4) even the term &quot;number&quot; is questionable since it is not clear how these so-called numbers can have anything to do with quantity (but then, complex numbers arguably also do not express quantity, or a single quantity); 5) no attempt to formally define what a decadic number is made; this so-called decadic number appears to be a mapping from positive integers to the set of digits 0-9, to be interpreted from right to left; 6) e.g. &quot;Addition of the decadic numbers is the same as that of the integers&quot; is clearly untrue: integers are finite discrete quantities; ditto for &quot;Multiplication works the same way in the decadic numbers as in the integers&quot;.

Perhaps this can be salvaged rather than moved out of mainspace. The first thing to do is add external sources dealing with the concept or state that this is original invention; and then, address the issues. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:30, 26 March 2024 (UTC)

:As with [[Surreal numbers]] the choice is between userspace and a subspace where users could be encouraged to cooperate. Unlike Surreal numbers, I am unaware of any application in physics for this topic. The ideal place would be [[Discrete mathematics/Number theory]] because the Olympiads is a high school thing. I will contact the author about both pages--[[Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:29, 26 March 2024 (UTC)
:: If the page should stay in mainspace, I see no reason why it could not stay at [[Decadic numbers]]; I don't see moving it around in mainspace as an improvement. But my position as explained above is that it is not fit for mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:07, 26 March 2024 (UTC)
:::Decadic numbers and Surreal numbers have enough that they should be parallel subpages of the same page. I have suggested to the author that they should either create a top page, or find a top page and group these resources together.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:54, 29 March 2024 (UTC)

== [[Rational numbers/Introduction]] ==
The page does not do anything that Wikipedia does not do better: [[Wikipedia: Rational number]]. The page contains unfilled tables that seemed to be intended to explain something, but since they are empty, explain nothing. The page has no further reading, revealing no attempt to find best complementary sources online, probably of much higher quality. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:20, 26 March 2024 (UTC)
:Now I see why you were kicked off Wiktionary. Wikiversity has a long and established tradition of allowing student efforts.  This page is no worse that [[Student Projects/Major rivers in India]], a page which I randomly selected from [[Student Projects]].  I am trying to recruit students to contribute to Wikiversity.  Until the Wikiversity community changes its mind about allowing student projects, I will continue with that quest. I will change the template so as to not discourage a person clearly interested in teaching mathematics, and I want you to refrain from placing rfd templates on student efforts.  Use {{tl|subpagify}} instead.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:24, 26 March 2024 (UTC)
:: I was blocked in the English Wiktionary for &quot;racism&quot; and more. In the English Wiktionary, I often defended pages nominated for deletion  and rather rarely nominated anything for deletion. The English Wiktionary has almost no useless pages and is the 2nd most often visited project after Wikipedia. By contrast, the English Wikiversity has very few useful pages, a state of affairs that I am trying to turn around, step by step, following processes and guidelines that I did nothing to establish: [[WV:RFD]] and [[WV:Deletions]]. That is as far as persons go (ad hominem); as far as process, I hoped here to have a discussion with editors about whether this nearly useless page ([[Rational numbers/Introduction]]) should be moved out of the mainspace, and unless consensus developed for my position, I stand no chance to prevail. [[Rational numbers/Introduction]]  is not a &quot;student project&quot; in any sense of &quot;project&quot; but rather example of all-too-typical junk. Again, I do not decide, others do with me being only a single voice/input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:52, 26 March 2024 (UTC)
:::Now you are on the right track! Wikiversity might be in a transition period between allowing all sorts of pages, to morphing into a selective institution.  But the process has to change from the top-down, not from the bottom by deleting one page at a time. When I say &quot;top&quot;, I am referring not to the administrators, but to the community at large. At present, RFD has nothing near the quorum required to implement the changes you (and others) are seeking. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:05, 26 March 2024 (UTC)
:::: The only reasonable way going forward, to my mind anyway, is to follow [[WV:Deletions]] and not worry about the precedent of its countless violations. Since, should we take e.g. [[Relation between Electricity And Magnetism]], existing since 2011‎, as an example of a page to be kept, then we must keep nearly everything. There are too many pages like that, and therefore, if we take their aggregate as a binding precedent to follow, we end up in trouble, unable to delete junk. It seems only fair to proceed according [[WV:Deletions]], especially when using RFD process which gives potential opposition enough time to object. Such a procedure violates neither established guidelines nor processes; if it &quot;violates&quot; anything, then preexisting extreme lenience/tolerance toward junk, lenience that, as far as I know, was never codified into a guideline. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:29, 26 March 2024 (UTC)
:::::No. Please don't use this page as an agenda for reforming Wikiversity. Go to the Colloquium or write an essay. Having said that, I did delete [[Creating Relation between Electricity And Magnetism]] because that follows both guidelines and established practice.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:50, 26 March 2024 (UTC)
:::::Actually, lenience is given an advantage when pages are up for deletion (See [[Special:Permalink/2615245#Wikipedia's_deletion_policy]] for evidence that deletion requires somewhat of a super-majority.) But you are not calling for deletion of low quality pages. Instead you want them out of mainspace.  We have room for compromise. But, as I said before: RFD is not the place to discuss this. If you want, I could take &quot;Wikiversity:What-goes-where 2024&quot; out of my user-space and we could discuss it there.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:18, 26 March 2024 (UTC)

==[[Student Projects/Major rivers in India]]==
{{Archive top|Closing with administrative decision to keep (snowball clause) --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:14, 29 March 2024 (UTC) -premature closure reverted. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:34, 30 March 2024 (UTC)}}

This page fails [[WV:Verifiability]], for one: surely the author cannot know these statements without consulting a source, but no source (zero) is provided. Thus, the author did nothing to meet a verification standard. The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. The page uses almost no wiki features, except for boldface, so the author did not practice wiki editing either. I would have used speedy nomination, but since I expect some opposition, I go for RFD. ''This shall be my last post to this RFD nomination''; I defer to the collective of other editors for the decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
* My vote: '''Move to draft archive''' or '''Delete'''; I prefer non-deletion since then we will be able to point to this as an example of a page that has no business being in the mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
*I moved it to [[Draft:Archive/2024/Student Projects/Major rivers in India]]-and then I moved it back-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)

This topic is closed due to the [[w:simple:Wikipedia:Snowball act|''Snowball clause'']]. For more information, see {{Permalink|2617055}}-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)
: A transparent link to what above is not a Wikiversity guideline/policy: [[w:simple:Wikipedia:Snowball act]]. It says &quot;stop things which don't have a snowball's chance in hell of passing&quot;. To my mind, this is an out-of-process premature closure, but indeed, in the current Wikiversity climate, I do not seem to have &quot;a snowball's chance in hell of&quot; ensuring proper process administration. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:45, 30 March 2024 (UTC)
::The snowball clause refers to the selective deletion of on page out of 300(?) pages with the same problem.  A proposal to remove '''all''' unsourced pages in [[Student Projects]] would be a new topic and that would require a new RFD proposal, as stated in {{tl|Archive top}} 
:: Also, [[Major rivers in India]] is a subset of the bigger problem at [[Student Projects]]. It would have taken you less time to add a new topic to RFD on [[Student Projects]], than it did for me to revert my closure of this topic.  {{tl|Archive top}} instructed you to open a new project.  By inserting text into the closed topic, you obligated me to unclose it.  I think you are deliberately trying to make things difficult for me.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:08, 30 March 2024 (UTC)
::: Maybe I should have numbered the reasons for deletion. You are right that 1) a complete lack of sourcing alone would probably be not grounds enough for deletion. But there is 2) The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. Wikiversity is not a duplicate of Wikipedia (of [[W: List of major rivers of India]]); it is especially not a bad duplicate of Wikipedia. If the page was someone's half-decent attempt to write a sourced encyclopedic article, I would have probably let it be, but as it stands, this text is not worth anyone's ''reading'' time, and if it was merely an exercise in writing, it should have stayed on the local hard drive. I feel I am kind enough to this text and its author in agreeing that this can be ''moved to draft archive''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:47, 1 April 2024 (UTC)
::: Near all RFD nominations are ''selective'' in that there nearly always exist many other pages with the same or similar problem that were not yet nominated. Once multiple RFDs confirm that the problem is indeed deletion-worthy/worthy of moving out of mainspace, we may even use speedy deletion nomination, given Wikiversity's traditional RFD-phobia. (I am happy to use RFD, but I go along with WV RFD-phobia and use speedy delete as far as possible, which I feel is administratively not so nice.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:54, 1 April 2024 (UTC)
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== [[Portal:Complex Systems Digital Campus/E-Laboratory on complex computational ecosystems/Members of the ECCE e-lab]] ==

I noticed a recent edit in the archives and stumbled upon an unanswered question by [[user:MGA73]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:29, 30 March 2024 (UTC)
: The linked page shows a list of laboratory members and their photo portraits (photos of faces). Such a thing does not seem to be particularly educational, and no big loss ensues by deletion. On the other hand, if this group of people wants to use Wikiversity to contribute research or educational material, this kind of page could be kindly tolerated. I do not really know what to do here. What is the precedent or similar previous RFD cases? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:06, 1 April 2024 (UTC)
::There is also a copyright problem and possibly a privacy issue.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:04, 1 April 2024 (UTC)
::: The page was created by [[User:Collet]] = Pierre Collet, who, believing the page, is one of three representatives of the group. Presumably, if these people did not want to be so published, they would not have agreed to Pierre's creating the page? Therefore, as for ''privacy'', should we assume a problem unless some of the members depicted contacts us, or should we rather assume Pierre Collet knew what he was doing? Pierre Collet's last edit was on 5 July 2021. Many of the images were uploaded by [[User:Pallamidessi]] in 2014, per [[Special:Contributions/Pallamidessi]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:48, 1 April 2024 (UTC)
:::: I am OK with '''keeping it as is'''.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:44, 12 April 2024 (UTC)

== [[Module:No globals]] ==
{{archive top|Deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:36, 12 April 2024 (UTC)}}
Replaced by the [[mw:Extension:Scribunto/Lua_reference_manual#strict|strict library]] of [[mw:Extension:Scribunto|Scribunto extension]]. --[[User:Liuxinyu970226|Liuxinyu970226]] ([[User talk:Liuxinyu970226|discuss]] • [[Special:Contributions/Liuxinyu970226|contribs]]) 11:40, 6 April 2024 (UTC)
:{{support}} deletion of unused and deprecated Module. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:26, 12 April 2024 (UTC)
:{{done}} Module deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:21, 12 April 2024 (UTC)
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== Unused files (user uploaded 1 file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 1 file only)]]. There are 115 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason.

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott and Katluvdogs was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded 1 file. So it is most likely not users that were very active on Wikiversity.

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:27, 29 April 2024 (UTC)

:Many of them are not useful for any real purpose. I'll chip away at some of these. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:48, 29 April 2024 (UTC)
::It was actually pretty easy to go thru most of these as they are 1.) clearly not useful, 2.) unused, or 3.) already exported to Commons. A substantial majority has been deleted. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:11, 29 April 2024 (UTC)
:::Did them all. Mostly lo-rez selfies, images of text, and diagrams or equations that related to nothing, plus a few screenplays. None of them were in use locally, a handf
::: were already on Commons and I exported some as well.l —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:29, 29 April 2024 (UTC)

== [[Template:Cc-by-nd-3.0]] and [[:Category:CC-BY-ND-3.0]] ==
ND is not a valid license on Wikiversity and there are no pages/files using the license so I suggest to delete the template and the category. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:02, 29 April 2024 (UTC)

:{{done}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:43, 29 April 2024 (UTC)

== Unused files (user uploaded 2-5 free file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 2-5 free file only)]]. There are 81 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason. 

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott, Katluvdogs and [[#Unused files (user uploaded 1 file only)]] was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded a few free files so the files are most likely not a part of a bigger set of files. 

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future.

There are some pdf-files among. Commons does usually not value pdf-files unless they are scans of old books for example. So I do not think we should move those files unless there is a good reason to do so. 

One of the files is called &quot;[[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]]&quot; so that would fit in [[:Category:WikiJournal]]. However it is also called &quot;preprint&quot; so I'm not sure if it is the final edition. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:32, 1 May 2024 (UTC)

: I removed [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] from category and added to [[:Category:WikiJournal Preprints]] to keep it per comment [[Special:Diff/2624512]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:41, 2 May 2024 (UTC)</text>
      <sha1>anc2945xyxrqm6w8unasvmd3mj1chlt</sha1>
    </revision>
    <revision>
      <id>2624894</id>
      <parentid>2624766</parentid>
      <timestamp>2024-05-03T03:59:21Z</timestamp>
      <contributor>
        <username>OhanaUnited</username>
        <id>18921</id>
      </contributor>
      <comment>/* Unused files uploaded by PCano */ example was an abandoned draft</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="66895" xml:space="preserve">{{/header}}
[[Category:Wikiversity deletion]]

'''&lt;big&gt;&lt;big&gt;&lt;big&gt;Deletion requests&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;'''

If an article should be deleted and does not meet [[WV:SPEEDY|speedy deletion criteria]], please list it here. Include the title and reason for deletion. If it meets speedy deletion criteria, just tag the resource with {{tlx|Delete|reason}} rather than opening a deletion discussion here.

If an article has been deleted, and you would like it undeleted, please list it here. Please try to give as close to the title as possible, and list your reasons for why it should be restored. The first line after the header should be: '''Undeletion requested'''

__TOC__

== Unused files uploaded by PCano ==
I suggest to delete the 287 unused files listed in [[:Category:Files uploaded by PCano - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2011 so it is unlikely the files will ever be used. The files seems to be a part of a set of data. I do not know if the set is complete. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:17, 26 February 2024 (UTC)
:I don't know the details, but sometimes the WikiJournals process the copyright differently. Has anybody checked with them about these files? If not, I would be happy to do the deed.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:34, 27 February 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]] I have not checked with WikiJournals. I was not thinking about copyright but if we are sure the files are correct and if they are of use to anyone? --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 27 February 2024 (UTC)
:::As I recall, files that are imbedded in pdf files are don't show up as being used.  I don't know why the WikiJournal would care,   the wikitext but want the pdf and raw files (wouldn't make any sense.) But the value of the Wikijournals is such that somebody needs to double check.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:52, 27 February 2024 (UTC)
:::: If the files are really embedded in a pdf (not linked), they are part of the pdf, and even if the files get deleted, the content is still in the pdf. What are examples of pdfs produced by Wikijurnals? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:20, 27 February 2024 (UTC)
* '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider. For the record, the files seem to be in the public domain, and many of them are for &quot;HLA allele distribution&quot;; &quot;Source: HumImmunol 2008&quot;. A selection of concerned file names: [[:File:2005 ASHI Poster 48 PCano.pdf]], [[:File:A-0101.gif]], [[:File:A-0102.gif]], [[:File:A-0103.gif]], [[:File:A-0201.gif]], [[:File:A-0202.gif]], [[:File:A-0203.gif]], [[:File:A-0204.gif]], [[:File:A-0205.gif]], [[:File:A-0206.gif]], [[:File:A-0207.gif]], [[:File:A-0208.gif]]. I randomly checked a couple of these files and they were uploaded in years 2010 and 2011. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:46, 27 February 2024 (UTC)
*Just to be safe, somebody needs to contact the WikiJournal.  This a a dormant author.  Right now my biggest problem is an active author.  I need to get an active author, [[User_talk:Saltrabook#Organizing_your_contributions|Saltrabook]], to put all their work under a single subpage before they become a bigger problem.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:29, 27 February 2024 (UTC)
*: No hurry here, AFAICT. This RFD can be opened for weeks and that is no big problem. And there are also other admins. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:50, 27 February 2024 (UTC)
*:: I made a comment at [[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]. Lets see if anyone join the discussion. And I agree that the discussion can be open for weeks. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:23, 27 February 2024 (UTC)
*:::''[[Talk:WikiJournal_User_Group#Notice_about_proposed_deletion]]'' has gone unnoticed for a month. What next?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:13, 28 March 2024 (UTC)
*:::: Delete :-) In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2608383 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:18, 28 March 2024 (UTC)
: More for the record and about the question where these files were probably used: The uploader [[User:PCano]] (Pedro Cano, M.D., M.B.A. MD Anderson Cancer Center, HLA Typing Laboratory, Houston, TX ) created [[Genetics/Human Leukocyte Antigen]] (originally under the title [[HLA]], moved to [[Genetics/Human Leukocyte Antigen]] in April 2017), which was much later (in December 2022) deleted as per [[Wikiversity:Requests for Deletion/Archives/18#Subpages of Genetics/Human Leukocyte Antigen]]. Deleting the files used there seems to be a natural follow-up on that deletion decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:23, 30 March 2024 (UTC)
::{{re|Guy vandegrift}} Unless you still worry about the WikiJournals I think you can delete the files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:54, 30 April 2024 (UTC)
:::I have a meeting with the WikJournal of Science tomorrow and I will bring it up.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:57, 30 April 2024 (UTC)
::::{{ping|MGA73|Dan Polansky}} I just talked to the WikiJournal editors and they have no problem with deleting these files.  Moreover, they have no problem with deleting any unused files, with one exception: They would prefer that we not delete pdf files that are marked as preprints, without first contacting them.  These preprint pdf files are easily identified with the standard WikiJournal preprint headers.  Apparently, they keep a record of all preprints and would need to create another depository for them if the Wikiversity community decides it doesn't want to host them.  Their policy is to post the preprint pdf files only if the article is submitted for publication.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:30, 1 May 2024 (UTC)
:::::{{re|Guy vandegrift}} Thank you. I added [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] to [[:Category:WikiJournal Preprints]] to remove it from deletion suggestion [[#Unused_files_(user_uploaded_2-5_free_file_only)]]. Perhaps some one can find the right category for it? Also It could be a good idea to make sure that all the WikJournal files are categorized somewhere in [[:Category:WikiJournal]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:44, 2 May 2024 (UTC)
::::::{{re|Mikael Häggström|Evolution and evolvability|OhanaUnited}} Have I correctly conveyed the wishes of the [[WikiJournal User Group/Editors|WJ editors]] in this regard?[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:32, 2 May 2024 (UTC)
:::::::Sounds good to me, thanks! [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 12:13, 2 May 2024 (UTC)
:::::::: Great! And as info I can tell that I made by bot add all files that seems to be related in any way to [[:Category:WikiJournal]]. For example if the word WikiJournal is used on the file page or the file is used on a page with WikiJournal in the title. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:16, 2 May 2024 (UTC)
:::::::::While this [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] file is a preprint within WikiJournal, the author never moved the PDF onto an actual preprint page. Judging from this author's [https://guc.toolforge.org/?by=date&amp;user=PARTHASARATHI.N global contributions], it's safe to say that the author abandoned the draft 4 years ago. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 03:59, 3 May 2024 (UTC)

== Archiving of Invalid fair use by User:Marshallsumter ==
* ''See [[Wikiversity:Requests for Deletion/Archives/21]]''

This space is for any unfinished business from that discussion.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 07:53, 29 February 2024 (UTC)
: Can be closed and archived, I guess. If anyone figures out a new task in the area of &quot;Invalid fair use by User:Marshallsumter&quot;, they can open a new RFD nomination as and when they do so. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:26, 30 March 2024 (UTC)
:: The problem is that the task (as mentioned in [[Wikiversity:Requests_for_Deletion/Archives/20#Pervasive copyright violations by User%3AMarshallsumter]]) is to check all the files uploaded by User:Marshallsumter and check if they meet the criteria for fair use. Sadly it is 1,151 files so I doubt anyone will spend the time on that. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:01, 30 March 2024 (UTC)
::: I tend to support preemptively deleting all files (not pages) uploaded by User:Marshallsumter. The fact that many of the files uploaded by him were determined not to meet Wikiversity criteria for fair use should be grounds enough. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:12, 30 March 2024 (UTC)
::::I thought we deleted all his files and userfied all his pages.  Apparently I was wrong: [[:File:Earth Shells to Scale.png]] // [[Earth/Geognosy/Quiz]] // [[Earth/Geognosy]]. When I deleted his images, I went to a page (category?) that someone else created. ... [https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 See also: This List].  Apparently this user spend all day long uploading files and putting them into pages he/she created. ... {{Ping|AP295}} This is why I don't bother with a couple of nutcase articles in [[Physics/Essays]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:55, 30 March 2024 (UTC)
::::: For anyone's interest, the upload list is visible at [[Special:ListFiles/Marshallsumter]]; a single-page view is at https://en.wikiversity.org/w/index.php?title=Special:ListFiles&amp;limit=1160&amp;user=Marshallsumter. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:09, 1 April 2024 (UTC)
{{outdent}} The abuse of the fair use doctrine by this former participant is so egregious that I fully support nuking all image uploads. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:25, 4 April 2024 (UTC)

:And I presume all pages by same participant that contain these images?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:09, 4 April 2024 (UTC)

::Any pages that have copious copyvio images should be deleted, along with the images. If there are pages without image violation they should be userfied. I doubt there are very many resources that have relevant learning content without copyvio. So, that leaves the resource pages open to deletion - which I support. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:44, 12 April 2024 (UTC)

=== Mixed discussion related to User:Marshallsumter and other topics ===
(Moved from [[Wikiversity:Requests_for_Deletion/Archives/22#User pages created as part of Computer Essentials (ICNS 141)]] --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC))
::{{ping|MGA73}} While I have your attention, I am confused about two lists that I compiled from various requests on RFD:
::#[https://en.wikiversity.org/w/index.php?title=Special:Contributions&amp;end=&amp;namespace=6&amp;start=&amp;tagfilter=&amp;target=Marshallsumter&amp;offset=&amp;limit=500 &gt;1500 Marshallsumter files]: ''Why we deleting Marshallsumter images?''
::#[[Draft:Original research/Literature]] &amp; [[Dominant group/Literature]] ''Marshallsumter sometimes delves into the &quot;soft&quot; (unscientific) subjects like literature where personal taste becomes important.  I see no reason to delete or even read them.''
::#{{Permalink|2608383|287 PCano files}} ''I believe these are being deleted because they are unused, yes?''
::#I am not very skilled at uploading files to commons that I did not create (most of my contributions need only attribution to other files on commons.)  I uploaded three files from the [[w:Library of Congress|loc]], and it was a time-consuming learning experience.  Is there someone else who can do it?  Perhaps I could watch till I got the hang of it.
::#After writing this I found {{Permalink|2497946#Exemption_Doctrine_Policy}}, which answers a lot of my questions.
::#I find this page a bit cluttered, but can live with it. If you want a general archiving and cleanup-just ask.
::--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:59, 2 April 2024 (UTC)
{{ping|Guy vandegrift}} Hello!
# Many of the files uploaded by Marshallsumter did not meet the requirements of fair use (violating the Exemption Doctrine Policy). I think all &quot;the easy files&quot; are deleted now. So to clean up the rest we either need hard work or a brute descision to delete everything just to be safe.
# I do not think I suggested to delete those 2 pages?
# Yes because they are unused.
# If you mean move files from here to Commons it is very easy: just click the tab &quot;Export to Wikimedia Commons&quot;. If you mean files you found on the Internet it is more tricky. You need to add the relevant information manually and more important add a source. If you found a website with hundreds or thousands of good files it may be possible to do with a bot (see [[:c:Commons:Batch uploading]]).
# Great :-)
# I can live with it too.
--[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:36, 9 April 2024 (UTC)

:On #1, I am happy with the brute decision if you are.  It's the uploader's responsibility to document the copyright.  Recently Mu301 and I &quot;rescued&quot; some high-quality photos on a high-quality resource.  But that was an exceptional case. Regarding #4, is (or should it be) our policy to move all Wikiversity files to Commons that are not fair use?  My problem with that is we sponsor some pretty low-quality stuff.  For example, instructors sometimes use Wikiversity for student submissions, and we can't delete those files until the course is over (in fact, we have no policy on deleting course-affiliated student submissions.) What do we do if the main page is a high-quality course, but some of the student submissions have no educational value?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 10 April 2024 (UTC)

::{{ping|Guy vandegrift}} I have no problem if everything is deleted in #1. And I also have no problems if course-affiliated student submissions are deleted after some time (#4). But I think both should be discussed on separate topics (perhaps just move the content to [[#Archiving_of_Invalid_fair_use_by_User:Marshallsumter]]). --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:41, 10 April 2024 (UTC)
:::I have been on Wikiversity for more than 10 years, most of the time not paying attention to such things, but I am unaware of any policy that calls for the routine deletion of student efforts that were created as part of an established course.  If no decision has ever been made to routinely delete student efforts, we need to make sure the entire community is on board with any change in policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 10 April 2024 (UTC)
::::Yes I agree. Deleting student efforts that were created as part of an established course needs a new discussion and concensus.
::::Except if it is a copyvio then it should be deleted. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:15, 12 April 2024 (UTC)

=== Deleting ALL non-free uploads by User:Marshallsumter ===
Okay so it seems everyone agree that files that violates Wikiversity criteria for fair use should be deleted - not a big surprise :-D

The big question is if files should be checked one by one or if they should all be deleted. I noticed that some users more or less support to delete all non-free files.

I therefore have 2 questions:
# Do you agree to delete all non-free files?
# Would you like to try to save any of the files and if yes should all the files be put on a list or in a category or how do you propose to make that possible?

Ping [[User:Guy vandegrift]], [[User:Dan Polansky]], [[User:Mu301]], [[User:Koavf]], [[User:Omphalographer]], [[User:Dave Braunschweig]], [[User:AP295]] and [[User:MathXplore]] that was involved in discussions recently. Sorry if I missed anyone and if you do not want to join this time thats of course okay. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:51, 26 April 2024 (UTC)

:#Yes
:#No
:—[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:52, 26 April 2024 (UTC)
::Also yes to 1 and no to 2, with the understanding that this policy only applies to MS because of the large volume of images involved.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:29, 27 April 2024 (UTC)
::: Correct and also because MS had a lot of bad files. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:52, 27 April 2024 (UTC)

:#yes, all non-free files should be deleted, prejudiced.
:#no, I don't believe that there is anything worth saving, in this batch from MS.
::--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 21:30, 27 April 2024 (UTC)
:: Same as Justin, Guy and mikeu: delete all Marshall Sumter-uploaded non-free files/uploads. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:20, 29 April 2024 (UTC)

== Unused files uploaded by Katluvdogs ==
{{Archive top|Consensus to delete all files. All are deleted.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:05, 12 April 2024 (UTC)}}
I suggest to delete the 137 unused files listed in [[:Category:Files uploaded by Katluvdogs - unused]]. A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about files uploaded by Robert Elliott was closed as delete above. Uploader have not been actice since 2009 so it is unlikely the files will ever be used. The files seems to be class notes but in order for the files to be usable they have to be categorized. Also it seems many are questions and questions are good but there should also be answers somewhere in order for it to be educational. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 10:35, 3 March 2024 (UTC)
* (Copying from elsewhere) '''Leaning toward delete''': since the files are unused, there seems to be no harm in deleting them. If someone presents arguments why they should be kept, I may reconsider.&lt;br/&gt;For the record, some of the files were referenced from [https://en.wikiversity.org/w/index.php?title=User:Katluvdogs/Ms.Puskarz:Class_Notes&amp;oldid=437947 this revision of User:Katluvdogs/Ms.Puskarz:Class_Notes], but the current revision of [[User:Katluvdogs/Ms.Puskarz:Class Notes]] states &quot;The website has been changed to: http://mspuskarzclassnotes.wikispaces.com/&quot;.&lt;br/&gt;On a minor note, pdfs are not a particularly good fit for a wiki, in my view.&lt;br/&gt;More for the record, a selection of the files being nominated for deletion: [[:File:3D cell model.pdf]], [[:File:Acid Rain Lab.pdf]], [[:File:Bio 16 and 17 hmwrk.pdf]], [[:File:BIO 18 H + SG.pdf]], [[:File:BIO 19 hmwrk and sg.pdf]], [[:File:BIO 20 hmwrk and sg.pdf]], [[:File:BIO 21 hmwrk.pdf]], [[:File:BIO 22 Hmwrk + sg.pdf]], [[:File:BIO 26 hmwrk + sg.pdf]], [[:File:BIO 27 hmwrk + sg.pdf]], [[:File:BIO April Calendar.pdf]], [[:File:BIO Bacteria infectious disease.pdf]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:56, 3 March 2024 (UTC)
:: In case anyone ever wonder which files it was they can see the files [https://en.wikiversity.org/w/index.php?title=User:MGA73/Sandbox&amp;oldid=2610073 in my sandbox history]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:20, 28 March 2024 (UTC)
:::'''Delete all files''' is my choice.  I see a 3-0 vote to delete, since Dan's vote was to delete if there are no objections (and nobody objected.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:12, 11 April 2024 (UTC)
:{{support}} deletion of all of these unused files. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:18, 12 April 2024 (UTC)
::{{done}} deleted all files in  Category:Files uploaded by Katluvdogs - unused--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:04, 12 April 2024 (UTC)
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==[[Ontosomose of Gender]]==
{{Archive top|Close with decision to move to [[Draft:Archive/2024/Ontosomose of Gender]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:14, 11 April 2024 (UTC)}}
I opened this RFD for a single purpose and that is: move this page created in 2007 by an anon to [[Draft:Archive/2024/Ontosomose of Gender]] rather than deleting it. It would be a pity to lose this little gem, quite possibly created as a joke. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:05, 6 March 2024 (UTC)  
:This is a good example of why we should just archive that which we do not understand.  It looks like gibberish to me, but [[w:Google Scholar|Google Scholar]] has [https://scholar.google.com/citations?user=eZgFGC0AAAAJ&amp;hl=de ''this article on him.'']  We may or may not be qualified to disagree with Google Scholar.  But we are certainly too small in number and to busy to look into everything Google Scholar deems worthy of mentioning. The suggestion that we move into [[Draft:Archive]] is seconded and {{done}}.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:06, 6 March 2024 (UTC)
:: @[[User:Guy vandegrift|Guy vandegrift]]: can you undelete [[Draft:Archive/2024/Ontosomose of Gender]], unless your intent is to actually have it deleted? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:37, 30 March 2024 (UTC)
:::I undeleted it. It was an accidental delete on my part. You may move it out of draft-archive.  Giving all editors the right to revert a move to draft-archive was my motive for creating [[Draft:Archive]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:22, 30 March 2024 (UTC)
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== [[OpenOffice.org]] ==

This one has me confused.  I used OpenOffice a long time ago, but grew tired of the advertising that came with the download.  The page looks good to me, but some subpages have been nominated for speedy deletion. What makes this case interesting is the [https://en.wikiversity.org/w/index.php?title=OpenOffice.org&amp;action=history ''history'']. Two high ranking WV administrators (Jtneill and Dave Braunschweig) worked hard to bring it up to speed, though I am sure neither currently objects to the project's deletion.  I drop their names so everybody believes me when I say that policy change is in the air.  Discuss it if you wish, or go ahead and make a vote so I can look for a consensus. It won't take much convincing to get me to move it to [[Draft:Archive/2024/OpenOffice.org]], especially if we leave a redirect. In fact, I will move with a redirect if anybody &quot;votes&quot; to move or delete.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:10, 7 March 2024 (UTC)
: I nominated [[OpenOffice.org/Writer]] and other subpages for speedy deletion. Looking at [[OpenOffice.org]], I do not see any saving grace either =&gt; delete, or move to userspace or move to draft archive. The page [[OpenOffice.org]] as it is does almost nothing to help one learn about OpenOffice.org; the few external links do not save it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:59, 9 March 2024 (UTC)
::I changed my vote to move relative material to WP because we don't need time-consuming solutions.  Will keep discussion open to permit others to perform the deed if they wish.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:52, 11 March 2024 (UTC)
::In the voting section I was asked why pages are safer in Draft:Archive-space than in Draft-space.  That got me thinking: Why do we have a policy that allows drafts to be deleted after 6 months?  Why not leave the effort in draft-space, with the understanding that anybody who want to improve the dormant draft can just blank it? This preserves the effort for whomever made it in the history of that draft? This will greatly reduce the number of pages that go into Draft:Archive.  I created Draft:Archive so that nobody's prior efforts would get lost.  The fewer pages I have to put there the better. We need a consensus to go into [[Wikiversity:Drafts]] and change that policy.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:18, 13 March 2024 (UTC)

===Voting on OpenOffice.org===
Please keep your vote, comment, and signature under 1 kB. Longer comments go in the section above.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:11, 7 March 2024 (UTC)

* '''Delete''' but move relevant material to [[w:OpenOffice]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:52, 7 March 2024 (UTC)
*'''Draft:Archive'''  (changed vote twice, now to match Dan's vote.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:20, 11 March 2024 (UTC)
*'''Draftify''' i propose this be moved to draft namespace.  or keep as is.  i see potential for this to spark creative ideas for other good faith Creative Commons content creation. Moving to draft namespace and potentially soft linking from an organizational archive page (ideally not as a sub-page) seems acceptable and sufficient. Willingness to not delete good faith contributions to the Creative Commons is greatly appreciated.  limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 17:44, 10 March 2024 (UTC) ... {{Ping|Michael Ten}} This page is safer in [[Draft:Archive/2024/OpenOffice.org]] than it is in [[Draft:OpenOffice.org]], so unless you object, I will consider your vote as a blessing to move it into Draft:Archive space.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:19, 11 March 2024 (UTC)
:: {{ping|Guy vandegrift}} I am not sure what you mean by &quot;This page is safer [...]&quot;  -- perhaps you mean it is likely likely to be effectively lost in the draft namespace or deleted from the draft namespace (?). I respect your views on that. I am happy enough that good faith contributions are moved to Draft namespace rather than deleted. I respect diversity of views and opinion about how Draft namespace could be best organized to be most collectively fruitful for the Creative Commons and this wiki. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:55, 13 March 2024 (UTC)
:::{{ping|Michael Ten}} According to [[Wikiversity:Drafts]], &quot;Resources which remain in the draft space for over 180 days (6 months) without being substantially edited may be deleted.&quot;. I do not like that policy, BTW.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:03, 13 March 2024 (UTC)
::::Interesting. Thank you for educating me on that. I agree with you; I do not think that is fruitful to the Creative Commons. You inspired [https://en.wikiversity.org/w/index.php?title=Wikiversity_talk%3ADrafts&amp;diff=2612156&amp;oldid=1998620 this suggestion]. Appreciated. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:11, 13 March 2024 (UTC)
* I wary of playing this &quot;!vote&quot; game, but I will: '''move to [[Draft:Archive]]''' or '''move to userpage''' or '''delete'''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:55, 11 March 2024 (UTC)
*I will move this to draft-archive because anybody can revert.  If nobody speaks in 10 days I will close and archive.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:22, 11 April 2024 (UTC)

==[[Metadata]]==
{{Archive top|Consens to draft-archive--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)}}
From my speedy deletion nomination: &quot;little to learn from here and the little that is here is from Wikipedia; no FR/EL&quot;. I have no objection to this being moved to user space or to [[Draft:Archive]]. Guideline: [[WV:Deletions]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:40, 9 March 2024 (UTC)
:As we decide what to do with &lt;code&gt;&lt;nowiki&gt;[[Metadata]]&lt;/nowiki&gt;&lt;/code&gt;, I assembled a choice of templates we might use in the future with such pages.  These templates use [[w:Help:Magic words|MAGIC WORDS]] that are connected to the current year and the page's location in namespace, and for that reason it is best to view the templates on a page that is actually up for deletion/pagemove. Two of the variations were designed by me to make it easier to copy/paste the new pagename (I also included the template's name to make it easier to for newbies to learn.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:59, 9 March 2024 (UTC)
:: I see only two templates as relevant: [[:Template:delete]] (speedy) and [[Template:rfd]]: non-speedy. I logged my disagreement to the template &quot;Draftify&quot; at [[Wikiversity:Colloquium#Template:Draftify]], which is what I think is the best place to discuss that template. I also created [[Wikiversity:Colloquium#Expanding WV:Deletions with Moving to Draft archive]] to codify what has recently been ongoing, namely that pages have been being moved to Draft archive instead of deleted; and I hope to get some supports there.
:: As for &quot;Metadata&quot;, the key decision is &quot;keep in mainspace&quot; vs. &quot;remove from mainspace&quot; and this is what this RFD is about.  --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:44, 10 March 2024 (UTC)
:::{{ping|Dan Polansky}} The problem with [[Wikiversity:Requests for Deletion]] is that is a [[w:Square peg in a round hole|&quot;round hole&quot; and the community is evolving towards &quot;square pegs&quot;]].  Meanwhile, I need a bottom line so I can look for sufficient consensus to act.o

===Voting on Metadata===
Change your vote as you wish. If you are not ready to vote, join the discussion directly above this &quot;voting section&quot;&quot;
*'''Delete, Draftify, or Userspace''' (&quot;vote&quot; cast on behalf of Dan Polansky by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:46, 11 March 2024 (UTC))
*'''Draftify IF the 6-month deletion rule is rescinded.'''  &lt;small&gt;Prior votes: From: Draft:Archive, to ''Delete, or Draftify'' in that order.&quot; &lt;/small&gt;--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:52, 28 March 2024 (UTC)
*''' Draftify''' (move to draft namespace) or keep as is. - Although moving to draft namespace seems acceptable and sufficient. limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:47, 13 March 2024 (UTC)
*{{done}} see [[Draft:Archive/2024/Metadata]][[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:11, 12 April 2024 (UTC)
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==[[Facilitation]]==
Trivial questions don't save what is a page with learning outcomes that are scarce ([[WV:Deletions]]]). I don't care whether this gets deleted, moved to userspace or moved to [[Draft:Archive]]. This was proposed for deletion in 2016 by Dave Braunschweig and was &quot;saved&quot; by adding questions that in my view are trivial and do not save the article. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:01, 10 March 2024 (UTC)
: '''Delete'''. I don't think the page achieves anything. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:50, 11 March 2024 (UTC)
:'''Draftify, pending vote to rescind the 6-month draftspace deletion rule''' (latest vote change)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:25, 23 March 2024 (UTC)
:'''Archive, Delete, or Userspace''' (roughly in that order: vote cast on behalf of [[User:Dan Polansky|Dan Polansky]] by [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:30, 11 March 2024 (UTC)&lt;small&gt;That's accurate. I guess I prefer Archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:57, 11 March 2024 (UTC)&lt;/small&gt;
:'''Draftify''' ('''Move to Draft namespace''') - I [https://en.wikiversity.org/w/index.php?title=Facilitation&amp;oldid=1633015 contributed to this page in good faith]. Deleting this page rather than preserving it somewhere will further decrease my motivations to contribute Creative Commons content to the Commons on this wiki, with the understanding that it is OK and considered a &quot;best practice&quot; to delete some good faith Creative Commons contributions on this wiki. A relevant rational may also be found [https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;diff=prev&amp;oldid=2611560 here]. Limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:43, 13 March 2024 (UTC)
:: The &quot;good faith&quot; talk is, in my view, entirely beside the point. Faith is not in question in deletion discussion, merely the aptness of the material for inclusion on a project, or inclusion in a specific namespace. For example, Wikiversity is not a repository of good-faith small children's creations or their analogues, or at least its mainspace is not. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:42, 16 March 2024 (UTC)
:: As an aside, the word you are looking for is &quot;rationale&quot;, not &quot;rational&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:37, 16 March 2024 (UTC)
:::{{ping|Dan Polansky}} I do not accept your premise that &quot;''Wikiversity is not a repository of (small children's creations)''&quot;. ... Also, there is a parallel discussion at [[Wikiversity_talk:Deletions#Proposed_modifications]], and it may remove most of the need for [[Draft:Archive]]. Michael Ten has pointed out that pages in draftspace could remain permanently. Looking back into the history, I discovered that I voted for the 180 limit.  I had forgotten all about that vote, but my own choice of wording jogged my memory: I voted for a 180 day limit because the decision to delete old drafts seemed like a foregone conclusion (Groupthink - who needs it!)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:51, 16 March 2024 (UTC)
:::: Well, then, from what does it follow that Wikiversity is such a repository? Which guideline, policy or scope statement? By small children I mean, say 0-6 years olds. Should e.g. scans of all pictures drawn by such children be uploadable as &quot;educational content&quot;? And if not pictures, should their first writings be uploadable? Why do they need publishing; does their local harddrive storage not serve the creative purpose enough? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:53, 16 March 2024 (UTC)
:::::I overstated my remark about children's work: For the most part, it belongs in userspace or draftspace. And, we need the parent's permission. But colleges teach courses in elementary education.  I once walked into such a course and somebody was reading a children's book to the entire class.  But we have no entrance requirements for Wikiversity, no minimum IQ is needed. Keep in mind that our differences are matters of personal taste (not factual reality.)  The question at hand at [[Wikiversity_talk:Deletions#Proposed_modifications]] is what requirements we wish to have for a page to reside indefinitely in draftspace.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:27, 16 March 2024 (UTC)
::::::I propose that we close this discussion with decision to delete, as author voted for that option.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:16, 4 April 2024 (UTC)

== [[HHF]] ==
{{Archive top|Moved to userspace--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:48, 1 April 2024 (UTC)}}
This page stands for &quot;High School Help Forum&quot;. It never became anything useful, it seems; it mostly contains pages that invite people to post but posts with actual content to learn from are missing. It has subpages that contain nothing useful, e.g. [[HHF/Physics/Introductory Physics]], [[HHF/Physics/Mechanics]], and [[HHF/Physics/Heat]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
* '''Move to Draft archive''' or '''move to user space''' or '''delete''', whatever is considered more appropriate. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 23 March 2024 (UTC)
*'''Draftify (pending vote to rescind 6-month draftspace deletion rule)''' Here's my problem: (1) Moving to draft space is not possible because the effort to allow unlimited presence in draft-space is stalled. (2) I don't want to move to draft-archive space because that is more time-consuming than moving to draft space. (3) Deletion is out because I strongly oppose, and I see no evidence of a community consensus to delete (as defined by Wikipedia and Wikitionary)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:23, 23 March 2024 (UTC)
*: There can very well be a 2/3-consensus to delete if one or two people join the discussion and say something like &quot;delete per [[WV:Deletions]]&quot;. Therefore, it does not seem true that deletion is out of question. It depends on who turns out and who decides to follow the actual guideline [[WV:Deletions]] as currently specified. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:22, 23 March 2024 (UTC)
*::I took a closer look at HHF and its 39(?) subpages. It's totally empty of content, but with an interesting use of wikitext. I could transfer three or so pages to [[Draft:Archive]] and send the rest to the author's userspace.  LIke with Marshallsumpter, it would have to be moved in about three parts because I can't even move that many pages in one operation.  That makes a 2-0 vote and I'm sure nobody else would object. See also [[User_talk:MathXplore#Question_about_soft_deletion]].... --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:54, 23 March 2024 (UTC)
-----
-----
;Update&lt;br&gt;
As this RFD page has gone dormant, I will probably draft-archive this page, but leave the templates intact (i.e. I won't dewikify it.)  It has occurred to me that since pages in draft-archive are organized by year, we can slowly dewikify after a few years of no edits. I will later post details on [[Wikiversity:What-goes-where 2024]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:39, 30 March 2024 (UTC)
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== [[Surreal numbers]] ==
{{Archive top|Page has a makeover buy me (it was fun!). Closing as &quot;keep in mainspace&quot;-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:14, 12 April 2024 (UTC)}}
1) Initially, this page made almost no sense to me; it did not explain what the &quot;{x|y}&quot; notation was supposed to mean. 2) However, from reading [[Wikipedia: Surreal numbers]], I see this notation is actually used. But then, the Wikiversity page has very little content and does not seem to do anything that the Wikipedia page does not do better. At a minimum, it should explain the notation. The page should only exist if it does something that Wikipedia does not do, e.g. by being more didactic or tutorial-like. 3) As always, I am fine with this being moved to user space or Draft archive. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:05, 26 March 2024 (UTC)
:Since it has only one author, the proper place would be userspace. It could also go into subpace as a student project in mathematics. I have [[Physics/Essays]], and it could easily go there.---[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:18, 26 March 2024 (UTC)
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== [[Decadic numbers]] ==
Arguably, this is not good enough for the mainspace; I have no objections to this being in the draft space or the userspace. Issues: 1) The page appears to be an original research but is not marked as such; 2) it introduces the term &quot;decadic number&quot; as an original terminological invention, as far as I can tell, but does not disclose this to be the case; 3) the term &quot;decadic number&quot; is unfortunate since what is meant is something like &quot;infinite decadic number&quot;; 4) even the term &quot;number&quot; is questionable since it is not clear how these so-called numbers can have anything to do with quantity (but then, complex numbers arguably also do not express quantity, or a single quantity); 5) no attempt to formally define what a decadic number is made; this so-called decadic number appears to be a mapping from positive integers to the set of digits 0-9, to be interpreted from right to left; 6) e.g. &quot;Addition of the decadic numbers is the same as that of the integers&quot; is clearly untrue: integers are finite discrete quantities; ditto for &quot;Multiplication works the same way in the decadic numbers as in the integers&quot;.

Perhaps this can be salvaged rather than moved out of mainspace. The first thing to do is add external sources dealing with the concept or state that this is original invention; and then, address the issues. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:30, 26 March 2024 (UTC)

:As with [[Surreal numbers]] the choice is between userspace and a subspace where users could be encouraged to cooperate. Unlike Surreal numbers, I am unaware of any application in physics for this topic. The ideal place would be [[Discrete mathematics/Number theory]] because the Olympiads is a high school thing. I will contact the author about both pages--[[Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:29, 26 March 2024 (UTC)
:: If the page should stay in mainspace, I see no reason why it could not stay at [[Decadic numbers]]; I don't see moving it around in mainspace as an improvement. But my position as explained above is that it is not fit for mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:07, 26 March 2024 (UTC)
:::Decadic numbers and Surreal numbers have enough that they should be parallel subpages of the same page. I have suggested to the author that they should either create a top page, or find a top page and group these resources together.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:54, 29 March 2024 (UTC)

== [[Rational numbers/Introduction]] ==
The page does not do anything that Wikipedia does not do better: [[Wikipedia: Rational number]]. The page contains unfilled tables that seemed to be intended to explain something, but since they are empty, explain nothing. The page has no further reading, revealing no attempt to find best complementary sources online, probably of much higher quality. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:20, 26 March 2024 (UTC)
:Now I see why you were kicked off Wiktionary. Wikiversity has a long and established tradition of allowing student efforts.  This page is no worse that [[Student Projects/Major rivers in India]], a page which I randomly selected from [[Student Projects]].  I am trying to recruit students to contribute to Wikiversity.  Until the Wikiversity community changes its mind about allowing student projects, I will continue with that quest. I will change the template so as to not discourage a person clearly interested in teaching mathematics, and I want you to refrain from placing rfd templates on student efforts.  Use {{tl|subpagify}} instead.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:24, 26 March 2024 (UTC)
:: I was blocked in the English Wiktionary for &quot;racism&quot; and more. In the English Wiktionary, I often defended pages nominated for deletion  and rather rarely nominated anything for deletion. The English Wiktionary has almost no useless pages and is the 2nd most often visited project after Wikipedia. By contrast, the English Wikiversity has very few useful pages, a state of affairs that I am trying to turn around, step by step, following processes and guidelines that I did nothing to establish: [[WV:RFD]] and [[WV:Deletions]]. That is as far as persons go (ad hominem); as far as process, I hoped here to have a discussion with editors about whether this nearly useless page ([[Rational numbers/Introduction]]) should be moved out of the mainspace, and unless consensus developed for my position, I stand no chance to prevail. [[Rational numbers/Introduction]]  is not a &quot;student project&quot; in any sense of &quot;project&quot; but rather example of all-too-typical junk. Again, I do not decide, others do with me being only a single voice/input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:52, 26 March 2024 (UTC)
:::Now you are on the right track! Wikiversity might be in a transition period between allowing all sorts of pages, to morphing into a selective institution.  But the process has to change from the top-down, not from the bottom by deleting one page at a time. When I say &quot;top&quot;, I am referring not to the administrators, but to the community at large. At present, RFD has nothing near the quorum required to implement the changes you (and others) are seeking. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:05, 26 March 2024 (UTC)
:::: The only reasonable way going forward, to my mind anyway, is to follow [[WV:Deletions]] and not worry about the precedent of its countless violations. Since, should we take e.g. [[Relation between Electricity And Magnetism]], existing since 2011‎, as an example of a page to be kept, then we must keep nearly everything. There are too many pages like that, and therefore, if we take their aggregate as a binding precedent to follow, we end up in trouble, unable to delete junk. It seems only fair to proceed according [[WV:Deletions]], especially when using RFD process which gives potential opposition enough time to object. Such a procedure violates neither established guidelines nor processes; if it &quot;violates&quot; anything, then preexisting extreme lenience/tolerance toward junk, lenience that, as far as I know, was never codified into a guideline. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:29, 26 March 2024 (UTC)
:::::No. Please don't use this page as an agenda for reforming Wikiversity. Go to the Colloquium or write an essay. Having said that, I did delete [[Creating Relation between Electricity And Magnetism]] because that follows both guidelines and established practice.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:50, 26 March 2024 (UTC)
:::::Actually, lenience is given an advantage when pages are up for deletion (See [[Special:Permalink/2615245#Wikipedia's_deletion_policy]] for evidence that deletion requires somewhat of a super-majority.) But you are not calling for deletion of low quality pages. Instead you want them out of mainspace.  We have room for compromise. But, as I said before: RFD is not the place to discuss this. If you want, I could take &quot;Wikiversity:What-goes-where 2024&quot; out of my user-space and we could discuss it there.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:18, 26 March 2024 (UTC)

==[[Student Projects/Major rivers in India]]==
{{Archive top|Closing with administrative decision to keep (snowball clause) --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:14, 29 March 2024 (UTC) -premature closure reverted. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:34, 30 March 2024 (UTC)}}

This page fails [[WV:Verifiability]], for one: surely the author cannot know these statements without consulting a source, but no source (zero) is provided. Thus, the author did nothing to meet a verification standard. The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. The page uses almost no wiki features, except for boldface, so the author did not practice wiki editing either. I would have used speedy nomination, but since I expect some opposition, I go for RFD. ''This shall be my last post to this RFD nomination''; I defer to the collective of other editors for the decision. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
* My vote: '''Move to draft archive''' or '''Delete'''; I prefer non-deletion since then we will be able to point to this as an example of a page that has no business being in the mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:32, 28 March 2024 (UTC)
*I moved it to [[Draft:Archive/2024/Student Projects/Major rivers in India]]-and then I moved it back-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)

This topic is closed due to the [[w:simple:Wikipedia:Snowball act|''Snowball clause'']]. For more information, see {{Permalink|2617055}}-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:11, 29 March 2024 (UTC)
: A transparent link to what above is not a Wikiversity guideline/policy: [[w:simple:Wikipedia:Snowball act]]. It says &quot;stop things which don't have a snowball's chance in hell of passing&quot;. To my mind, this is an out-of-process premature closure, but indeed, in the current Wikiversity climate, I do not seem to have &quot;a snowball's chance in hell of&quot; ensuring proper process administration. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:45, 30 March 2024 (UTC)
::The snowball clause refers to the selective deletion of on page out of 300(?) pages with the same problem.  A proposal to remove '''all''' unsourced pages in [[Student Projects]] would be a new topic and that would require a new RFD proposal, as stated in {{tl|Archive top}} 
:: Also, [[Major rivers in India]] is a subset of the bigger problem at [[Student Projects]]. It would have taken you less time to add a new topic to RFD on [[Student Projects]], than it did for me to revert my closure of this topic.  {{tl|Archive top}} instructed you to open a new project.  By inserting text into the closed topic, you obligated me to unclose it.  I think you are deliberately trying to make things difficult for me.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:08, 30 March 2024 (UTC)
::: Maybe I should have numbered the reasons for deletion. You are right that 1) a complete lack of sourcing alone would probably be not grounds enough for deletion. But there is 2) The reader does not learn anything they could not have learned in Wikipedia =&gt; no value for the reader. Wikiversity is not a duplicate of Wikipedia (of [[W: List of major rivers of India]]); it is especially not a bad duplicate of Wikipedia. If the page was someone's half-decent attempt to write a sourced encyclopedic article, I would have probably let it be, but as it stands, this text is not worth anyone's ''reading'' time, and if it was merely an exercise in writing, it should have stayed on the local hard drive. I feel I am kind enough to this text and its author in agreeing that this can be ''moved to draft archive''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:47, 1 April 2024 (UTC)
::: Near all RFD nominations are ''selective'' in that there nearly always exist many other pages with the same or similar problem that were not yet nominated. Once multiple RFDs confirm that the problem is indeed deletion-worthy/worthy of moving out of mainspace, we may even use speedy deletion nomination, given Wikiversity's traditional RFD-phobia. (I am happy to use RFD, but I go along with WV RFD-phobia and use speedy delete as far as possible, which I feel is administratively not so nice.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:54, 1 April 2024 (UTC)
{{archive bottom}}

== [[Portal:Complex Systems Digital Campus/E-Laboratory on complex computational ecosystems/Members of the ECCE e-lab]] ==

I noticed a recent edit in the archives and stumbled upon an unanswered question by [[user:MGA73]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:29, 30 March 2024 (UTC)
: The linked page shows a list of laboratory members and their photo portraits (photos of faces). Such a thing does not seem to be particularly educational, and no big loss ensues by deletion. On the other hand, if this group of people wants to use Wikiversity to contribute research or educational material, this kind of page could be kindly tolerated. I do not really know what to do here. What is the precedent or similar previous RFD cases? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:06, 1 April 2024 (UTC)
::There is also a copyright problem and possibly a privacy issue.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:04, 1 April 2024 (UTC)
::: The page was created by [[User:Collet]] = Pierre Collet, who, believing the page, is one of three representatives of the group. Presumably, if these people did not want to be so published, they would not have agreed to Pierre's creating the page? Therefore, as for ''privacy'', should we assume a problem unless some of the members depicted contacts us, or should we rather assume Pierre Collet knew what he was doing? Pierre Collet's last edit was on 5 July 2021. Many of the images were uploaded by [[User:Pallamidessi]] in 2014, per [[Special:Contributions/Pallamidessi]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:48, 1 April 2024 (UTC)
:::: I am OK with '''keeping it as is'''.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:44, 12 April 2024 (UTC)

== [[Module:No globals]] ==
{{archive top|Deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:36, 12 April 2024 (UTC)}}
Replaced by the [[mw:Extension:Scribunto/Lua_reference_manual#strict|strict library]] of [[mw:Extension:Scribunto|Scribunto extension]]. --[[User:Liuxinyu970226|Liuxinyu970226]] ([[User talk:Liuxinyu970226|discuss]] • [[Special:Contributions/Liuxinyu970226|contribs]]) 11:40, 6 April 2024 (UTC)
:{{support}} deletion of unused and deprecated Module. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:26, 12 April 2024 (UTC)
:{{done}} Module deleted. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:21, 12 April 2024 (UTC)
{{archive bottom}}

== Unused files (user uploaded 1 file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 1 file only)]]. There are 115 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason.

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott and Katluvdogs was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded 1 file. So it is most likely not users that were very active on Wikiversity.

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:27, 29 April 2024 (UTC)

:Many of them are not useful for any real purpose. I'll chip away at some of these. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:48, 29 April 2024 (UTC)
::It was actually pretty easy to go thru most of these as they are 1.) clearly not useful, 2.) unused, or 3.) already exported to Commons. A substantial majority has been deleted. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:11, 29 April 2024 (UTC)
:::Did them all. Mostly lo-rez selfies, images of text, and diagrams or equations that related to nothing, plus a few screenplays. None of them were in use locally, a handf
::: were already on Commons and I exported some as well.l —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:29, 29 April 2024 (UTC)

== [[Template:Cc-by-nd-3.0]] and [[:Category:CC-BY-ND-3.0]] ==
ND is not a valid license on Wikiversity and there are no pages/files using the license so I suggest to delete the template and the category. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 15:02, 29 April 2024 (UTC)

:{{done}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:43, 29 April 2024 (UTC)

== Unused files (user uploaded 2-5 free file only) ==
I suggest to delete the unused files in [[:Category:Unused files (user uploaded 2-5 free file only)]]. There are 81 files but a few are also in [[:Category:NowCommons]] and could be speedy deleted for that reason. 

A longer discussion about unused files in general can be seen at [[Wikiversity:Requests_for_Deletion/Archives/20#Thousands_of_unused_files]] and a similar discussion about unused files uploaded by Robert Elliott, Katluvdogs and [[#Unused files (user uploaded 1 file only)]] was closed as delete. Currently there is an open discussion for [[#Unused_files_uploaded_by_PCano]].

These files were uploaded by users that only uploaded a few free files so the files are most likely not a part of a bigger set of files. 

I made a comment at [[Wikiversity:Colloquium#Moving_free_files_to_Commons]] about moving files to Commons but I do not think these files look useful. If anyone think that one or more of the files should be kept they are welcome to move them to Commons so they can be put in relevant categories and hopefully be used for something in the future.

There are some pdf-files among. Commons does usually not value pdf-files unless they are scans of old books for example. So I do not think we should move those files unless there is a good reason to do so. 

One of the files is called &quot;[[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]]&quot; so that would fit in [[:Category:WikiJournal]]. However it is also called &quot;preprint&quot; so I'm not sure if it is the final edition. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 14:32, 1 May 2024 (UTC)

: I removed [[:File:WikiJournal Preprints COVID-19 ELIMINATION AND CELL DIFFERENTIATION - Wikiversity.pdf]] from category and added to [[:Category:WikiJournal Preprints]] to keep it per comment [[Special:Diff/2624512]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:41, 2 May 2024 (UTC)</text>
      <sha1>djrvkgr3w5uqv01a7i5it6d4rcl64jl</sha1>
    </revision>
  </page>
  <page>
    <title>List of extinct languages</title>
    <ns>0</ns>
    <id>3851</id>
    <revision>
      <id>2624710</id>
      <parentid>2210862</parentid>
      <timestamp>2024-05-02T16:03:22Z</timestamp>
      <contributor>
        <ip>173.12.99.51</ip>
      </contributor>
      <comment>/* Indo-European Languages */ Added livonian</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1183" xml:space="preserve">== Origin Uncertain ==
* [[Portal:Sumerian|Sumerian]] probably afroasiaotic
* Kassite
* Elamite
* Ancient Iberian (of Iberian Peninsula)
* Tartessian
* Aquitanian
* Etruscan 
* Lemnian
* Raetic
* Minoan (or Linear A of Crete)
* Hattic
* Meroitic

== Indo-European Languages ==
====Anatolian Languages (All Extinct)====
* Cuneiform Hittite
* Hieroglyphic Hittite
* Luwian
* Palaian
* Lydian
* Lycian
* Carian

====Indo-Iranian Languages====
* Avestan
* Ancient Persian (Cuneiform Persian)
* Pahlavi (Middle Persian)

====  Italic Languages====
* Latin
* Oscian
* Umbrian
* Venetic
* Lombardian
* Livonian

====  Ancient Languages of Balkans====
* Phrygian
* Macedonian (Not to be confused with modern Macedonian of Slavic Group)
* Thracian
* Illyrian
* Dacian

====  Celtic Languages====
* British
* Pictish

== Hurro-Urartian Languages ==
* Hurrian
* Urartian

== Afroasiatic Languages ==
* Akkadian
* Babylonian( scientists have uncovered some of the language and are speaking it)
* Aramaic (Assyrians are one of the few left)
* Ugaritic
* Phoenician
* [[Ancient Egyptian Language|Ancient Egyptian]]

[[Category:Lists]]
[[Category:Pages moved from Wikibooks]]
[[Category:Languages]]</text>
      <sha1>cwd304t5emfldvbof0f65j5cym8ojyp</sha1>
    </revision>
  </page>
  <page>
    <title>Conic sections</title>
    <ns>0</ns>
    <id>4990</id>
    <revision>
      <id>2624960</id>
      <parentid>2624025</parentid>
      <timestamp>2024-05-03T10:54:07Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Other resources */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40619" xml:space="preserve">'''Conic sections''' are curves created by the intersection of a plane and a cone.  There are six types of conic section: the circle, ellipse, hyperbola, parabola, a pair of intersecting straight lines and a single point.

All conics (as they are known) have at least two foci, although the two may coincide or one may be at infinity.  They may also be defined as the locus of a point moving between a point and a line, a '''directrix''', such that the ratio between the distances is constant. This ratio is known as &quot;e&quot;, or [[eccentricity]].

== Ellipses ==
[[Image:Ellipse Animation Small.gif|thumb|right|300px|Animation showing distance from the foci of an ellipse to several different points on the ellipse.]]

An ellipse is a locus where the sum of the distances to two foci is kept constant.  This sum is also equivalent to the major axis of the ellipse - the major axis being longer of the two lines of symmetry of the ellipse, running through both foci.  The eccentricity of an ellipse is less than one. 

In [[w:Cartesian coordinate system|Cartesian coordinates]], if an ellipse is centered at (h,k), the equation for the ellipse is

:&lt;math&gt;\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1&lt;/math&gt; &amp;nbsp;&amp;nbsp;&amp;nbsp;(equation 1)

The lengths of the major and minor axes (also known as the conjugate and transverse) are &quot;a&quot; and &quot;b&quot; respectively.

'''Exercise 1'''. Derive equation 1. &amp;nbsp;&amp;nbsp;&amp;nbsp;([[w:Ellipse/Proofs|hint]])

A circle circumscribed about the ellipse, touching at the two end points of the major axis, is known as the [[auxiliary circle]]. The [[latus rectum]] of an ellipse passes through the foci and is perpendicular to the major axis.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} - \frac{yb^2}{y_1} = a^2 - b^2&lt;/math&gt;

Likewise for the [[parametric coordinates]] of P, (a &lt;math&gt;\cos \alpha&lt;/math&gt;, b &lt;math&gt;\sin \alpha&lt;/math&gt;),

:&lt;math&gt;\frac{x\cos\alpha}{a} + \frac{y\sin\alpha}{b} = 1&lt;/math&gt;

== Properties of Ellipses ==

S and S' are typically regarded as the two foci of the ellipse. Where &lt;math&gt;a &gt; b&lt;/math&gt;, these become (ae, 0) and (-ae, 0) respectively. Where &lt;math&gt;a &lt; b&lt;/math&gt; these become (0, be) and (0, -be) respectively.

A point ''P'' on the ellipse will move about these two foci [[ut]]
&lt;math&gt;|PS + PS'| = 2a&lt;/math&gt;

Where a &gt; b, which is to say the Ellipse will have a major-axes parallel to the x-axis:

&lt;math&gt;b^2 = a^2(1-e^2)&lt;/math&gt;

The directrix will be: 
&lt;math&gt;x = \pm \frac{a}{e}&lt;/math&gt;

== Circles ==
A circle is a special type of the ellipse where the foci are the same point.

Hence, the equation becomes:

&lt;math&gt;x^2+y^2 = r^2&lt;/math&gt;

''Where 'r' represents the radius.''
And the circle is centered at the origin (0,0)

== Hyperbolas ==
A special case where the eccentricity of the conic shape is greater than one. 

Centered at the origin, Hyperbolas have the general equation:

:&lt;math&gt;\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1&lt;/math&gt;

A point ''P'' on will move about the two foci ut &lt;math&gt;|PS - PS'| = 2a&lt;/math&gt;

The equations for the tangent and normal to the hyperbola closely resemble that of the ellipse.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} + \frac{yb^2}{y_1} = a^2 + b^2&lt;/math&gt;

The directrixes (singular directrix) and foci of hyperbolas are the same as those of ellipses, namely directrixes of &lt;math&gt; x = \pm \frac{a}{e} &lt;/math&gt;   and foci of   &lt;math&gt; ( \pm ae, 0) &lt;/math&gt;

The [[asymptotes]] of a hyperbola lie at &lt;math&gt; y = \pm \frac {b}{a}x &lt;/math&gt;

== Rectangular Hyperbolas ==
Rectangular Hyperbolas are special cases of hyperbolas where the asymptotes are perpendicular. 
These have the general equation:

&lt;math&gt; xy = c &lt;/math&gt;

==Conic sections generally==
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the conic section, and also how to calculate the foci and directrices given the equation.
===Deriving the equation===
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
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Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
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===Implementation===

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

===Examples===
====Parabola====
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
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[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
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=====Gallery=====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
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====Ellipse====
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
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[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
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=====Gallery=====

{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
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====Hyperbola====
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of curve approaches shape of parabola.
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
=====Gallery=====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to X axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Reversing the process===
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain an unwanted value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
====Implementation====
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====More calculations====
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
====Implementation====
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===

====Parabola====
{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Ellipse====
{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Other resources==
*Should the contents of this Wikiversity page be merged into the related Wikibooks modules such as [[b:Conic Sections/Ellipse]]?

[[Category:Geometry]]
[[Category:Resources last modified in December 2012]]</text>
      <sha1>3b1dvmjh9mejht64y02mqn1kktct45q</sha1>
    </revision>
    <revision>
      <id>2624961</id>
      <parentid>2624960</parentid>
      <timestamp>2024-05-03T10:58:07Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Implementation */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="42193" xml:space="preserve">'''Conic sections''' are curves created by the intersection of a plane and a cone.  There are six types of conic section: the circle, ellipse, hyperbola, parabola, a pair of intersecting straight lines and a single point.

All conics (as they are known) have at least two foci, although the two may coincide or one may be at infinity.  They may also be defined as the locus of a point moving between a point and a line, a '''directrix''', such that the ratio between the distances is constant. This ratio is known as &quot;e&quot;, or [[eccentricity]].

== Ellipses ==
[[Image:Ellipse Animation Small.gif|thumb|right|300px|Animation showing distance from the foci of an ellipse to several different points on the ellipse.]]

An ellipse is a locus where the sum of the distances to two foci is kept constant.  This sum is also equivalent to the major axis of the ellipse - the major axis being longer of the two lines of symmetry of the ellipse, running through both foci.  The eccentricity of an ellipse is less than one. 

In [[w:Cartesian coordinate system|Cartesian coordinates]], if an ellipse is centered at (h,k), the equation for the ellipse is

:&lt;math&gt;\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1&lt;/math&gt; &amp;nbsp;&amp;nbsp;&amp;nbsp;(equation 1)

The lengths of the major and minor axes (also known as the conjugate and transverse) are &quot;a&quot; and &quot;b&quot; respectively.

'''Exercise 1'''. Derive equation 1. &amp;nbsp;&amp;nbsp;&amp;nbsp;([[w:Ellipse/Proofs|hint]])

A circle circumscribed about the ellipse, touching at the two end points of the major axis, is known as the [[auxiliary circle]]. The [[latus rectum]] of an ellipse passes through the foci and is perpendicular to the major axis.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} - \frac{yb^2}{y_1} = a^2 - b^2&lt;/math&gt;

Likewise for the [[parametric coordinates]] of P, (a &lt;math&gt;\cos \alpha&lt;/math&gt;, b &lt;math&gt;\sin \alpha&lt;/math&gt;),

:&lt;math&gt;\frac{x\cos\alpha}{a} + \frac{y\sin\alpha}{b} = 1&lt;/math&gt;

== Properties of Ellipses ==

S and S' are typically regarded as the two foci of the ellipse. Where &lt;math&gt;a &gt; b&lt;/math&gt;, these become (ae, 0) and (-ae, 0) respectively. Where &lt;math&gt;a &lt; b&lt;/math&gt; these become (0, be) and (0, -be) respectively.

A point ''P'' on the ellipse will move about these two foci [[ut]]
&lt;math&gt;|PS + PS'| = 2a&lt;/math&gt;

Where a &gt; b, which is to say the Ellipse will have a major-axes parallel to the x-axis:

&lt;math&gt;b^2 = a^2(1-e^2)&lt;/math&gt;

The directrix will be: 
&lt;math&gt;x = \pm \frac{a}{e}&lt;/math&gt;

== Circles ==
A circle is a special type of the ellipse where the foci are the same point.

Hence, the equation becomes:

&lt;math&gt;x^2+y^2 = r^2&lt;/math&gt;

''Where 'r' represents the radius.''
And the circle is centered at the origin (0,0)

== Hyperbolas ==
A special case where the eccentricity of the conic shape is greater than one. 

Centered at the origin, Hyperbolas have the general equation:

:&lt;math&gt;\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1&lt;/math&gt;

A point ''P'' on will move about the two foci ut &lt;math&gt;|PS - PS'| = 2a&lt;/math&gt;

The equations for the tangent and normal to the hyperbola closely resemble that of the ellipse.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} + \frac{yb^2}{y_1} = a^2 + b^2&lt;/math&gt;

The directrixes (singular directrix) and foci of hyperbolas are the same as those of ellipses, namely directrixes of &lt;math&gt; x = \pm \frac{a}{e} &lt;/math&gt;   and foci of   &lt;math&gt; ( \pm ae, 0) &lt;/math&gt;

The [[asymptotes]] of a hyperbola lie at &lt;math&gt; y = \pm \frac {b}{a}x &lt;/math&gt;

== Rectangular Hyperbolas ==
Rectangular Hyperbolas are special cases of hyperbolas where the asymptotes are perpendicular. 
These have the general equation:

&lt;math&gt; xy = c &lt;/math&gt;

==Conic sections generally==
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the conic section, and also how to calculate the foci and directrices given the equation.
===Deriving the equation===
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

===Implementation===

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

===Examples===
====Parabola====
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
=====Gallery=====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

====Ellipse====
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}

=====Gallery=====

{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
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====Hyperbola====
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
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[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
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[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of curve approaches shape of parabola.
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
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=====Gallery=====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to X axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
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===Reversing the process===
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
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Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain an unwanted value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
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====Implementation====
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&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
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====More calculations====
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The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
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====Implementation====
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&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
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===Examples===

====Parabola====
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[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
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====Ellipse====
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[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x = 17, slope = 5.
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Other resources==
*Should the contents of this Wikiversity page be merged into the related Wikibooks modules such as [[b:Conic Sections/Ellipse]]?

[[Category:Geometry]]
[[Category:Resources last modified in December 2012]]</text>
      <sha1>9v3dirnz67ltx4mnrygfngv1046blf9</sha1>
    </revision>
    <revision>
      <id>2624962</id>
      <parentid>2624961</parentid>
      <timestamp>2024-05-03T11:00:45Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* y = f(x) */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43951" xml:space="preserve">'''Conic sections''' are curves created by the intersection of a plane and a cone.  There are six types of conic section: the circle, ellipse, hyperbola, parabola, a pair of intersecting straight lines and a single point.

All conics (as they are known) have at least two foci, although the two may coincide or one may be at infinity.  They may also be defined as the locus of a point moving between a point and a line, a '''directrix''', such that the ratio between the distances is constant. This ratio is known as &quot;e&quot;, or [[eccentricity]].

== Ellipses ==
[[Image:Ellipse Animation Small.gif|thumb|right|300px|Animation showing distance from the foci of an ellipse to several different points on the ellipse.]]

An ellipse is a locus where the sum of the distances to two foci is kept constant.  This sum is also equivalent to the major axis of the ellipse - the major axis being longer of the two lines of symmetry of the ellipse, running through both foci.  The eccentricity of an ellipse is less than one. 

In [[w:Cartesian coordinate system|Cartesian coordinates]], if an ellipse is centered at (h,k), the equation for the ellipse is

:&lt;math&gt;\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1&lt;/math&gt; &amp;nbsp;&amp;nbsp;&amp;nbsp;(equation 1)

The lengths of the major and minor axes (also known as the conjugate and transverse) are &quot;a&quot; and &quot;b&quot; respectively.

'''Exercise 1'''. Derive equation 1. &amp;nbsp;&amp;nbsp;&amp;nbsp;([[w:Ellipse/Proofs|hint]])

A circle circumscribed about the ellipse, touching at the two end points of the major axis, is known as the [[auxiliary circle]]. The [[latus rectum]] of an ellipse passes through the foci and is perpendicular to the major axis.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} - \frac{yb^2}{y_1} = a^2 - b^2&lt;/math&gt;

Likewise for the [[parametric coordinates]] of P, (a &lt;math&gt;\cos \alpha&lt;/math&gt;, b &lt;math&gt;\sin \alpha&lt;/math&gt;),

:&lt;math&gt;\frac{x\cos\alpha}{a} + \frac{y\sin\alpha}{b} = 1&lt;/math&gt;

== Properties of Ellipses ==

S and S' are typically regarded as the two foci of the ellipse. Where &lt;math&gt;a &gt; b&lt;/math&gt;, these become (ae, 0) and (-ae, 0) respectively. Where &lt;math&gt;a &lt; b&lt;/math&gt; these become (0, be) and (0, -be) respectively.

A point ''P'' on the ellipse will move about these two foci [[ut]]
&lt;math&gt;|PS + PS'| = 2a&lt;/math&gt;

Where a &gt; b, which is to say the Ellipse will have a major-axes parallel to the x-axis:

&lt;math&gt;b^2 = a^2(1-e^2)&lt;/math&gt;

The directrix will be: 
&lt;math&gt;x = \pm \frac{a}{e}&lt;/math&gt;

== Circles ==
A circle is a special type of the ellipse where the foci are the same point.

Hence, the equation becomes:

&lt;math&gt;x^2+y^2 = r^2&lt;/math&gt;

''Where 'r' represents the radius.''
And the circle is centered at the origin (0,0)

== Hyperbolas ==
A special case where the eccentricity of the conic shape is greater than one. 

Centered at the origin, Hyperbolas have the general equation:

:&lt;math&gt;\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1&lt;/math&gt;

A point ''P'' on will move about the two foci ut &lt;math&gt;|PS - PS'| = 2a&lt;/math&gt;

The equations for the tangent and normal to the hyperbola closely resemble that of the ellipse.

From a point P(&lt;math&gt;x_1&lt;/math&gt;, &lt;math&gt;y_1&lt;/math&gt;) tangents will have the equation:

:&lt;math&gt;\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1&lt;/math&gt;

And normals:

:&lt;math&gt;\frac{xa^2}{x_1} + \frac{yb^2}{y_1} = a^2 + b^2&lt;/math&gt;

The directrixes (singular directrix) and foci of hyperbolas are the same as those of ellipses, namely directrixes of &lt;math&gt; x = \pm \frac{a}{e} &lt;/math&gt;   and foci of   &lt;math&gt; ( \pm ae, 0) &lt;/math&gt;

The [[asymptotes]] of a hyperbola lie at &lt;math&gt; y = \pm \frac {b}{a}x &lt;/math&gt;

== Rectangular Hyperbolas ==
Rectangular Hyperbolas are special cases of hyperbolas where the asymptotes are perpendicular. 
These have the general equation:

&lt;math&gt; xy = c &lt;/math&gt;

==Conic sections generally==
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the conic section, and also how to calculate the foci and directrices given the equation.
===Deriving the equation===
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

===Implementation===

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

===Examples===
====Parabola====
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus &lt;/br&gt;= distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
=====Gallery=====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

====Ellipse====
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}

=====Gallery=====

{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}

====Hyperbola====
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of curve approaches shape of parabola.
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
=====Gallery=====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to X axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Reversing the process===
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain an unwanted value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
====Implementation====
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====More calculations====
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
====Implementation====
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===

====Parabola====
{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Ellipse====
{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x = 17, slope = 5.
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
=====x = f(y)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Other resources==
*Should the contents of this Wikiversity page be merged into the related Wikibooks modules such as [[b:Conic Sections/Ellipse]]?

[[Category:Geometry]]
[[Category:Resources last modified in December 2012]]</text>
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    <title>Hegel</title>
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&lt; Back to the [[School:Philosophy|School of Philosophy]]

{{cquote|[I have seen the master composing with obscure and intricate signs, so that they could not be easily deciphered.] I often used to see him looking around anxiously as if in fear he might be understood. He was very fond of me, for he was sure I would never betray him. As a matter of fact, I then thought that he was very obsequious. Once, when I grew impatient with him for saying: 'All that is, is rational', he smiled strangely and remarked, 'it may also be said that all that is rational must be'. Then he looked about him hastily; but he was speedily reassured, for only Heinrich Beer had heard his words. (Heinrich Heine, cited in 
&lt;strong&gt;McCarney, J&lt;/strong&gt;: &lt;em&gt;Hegel on History&lt;/em&gt;, {{ISBN|0-415-11695-3}})}}

[[w:Georg_Wilhelm_Friedrich_Hegel|Hegel]] has become notorious for the incredible obscurity of his writing. I am reading the back cover of [[w:Peter Singer|Peter Singer's]] introduction to Hegel, where one can find a typical description of the non-specialist's impression.&lt;blockquote&gt;Hegel has become a stock example of an obscure philosopher — a name to conjure with, but not someone whose work can be read and understood. Yet his importance is universally acknowledged, and we are still living in an intellectual climate decisively influenced by his ideas. ({{ISBN|0-19-287564-7}})&lt;/blockquote&gt;
However, it is probably best to read something written by Hegel to form your own opinion about his prose. A good place to start is the first couple of pages of the ''Introduction'' in the ''[[w:Phenomenology of Spirit|Phenomenology of Spirit]]''.

Those who are not intimidated by the text face a further difficulty: Popular misconceptions or misleading caricatures of Hegel's work that abound even among academics unschooled in philosophy&lt;ref&gt;{{Cite web|url=http://articles.adsabs.harvard.edu//full/1992JHA....23..208C/0000208.000.html|title=Hegel and the Seven Planets|last=Craig|first=E|last2=Hoskin|first2=M.|date=1992-08-03|website=articles.adsabs.harvard.edu|archive-url=|archive-date=|access-date=2020-07-30}}&lt;/ref&gt;. Despite being conspicuously debunked, these prejudices unjustly generate antipathy for Hegel's works. Crucial to the work of dispelling these myths has been [https://https://en.wikipedia.org/wiki/Jon_Stewart_(philosopher) Jon Stewart's] ''The Hegel Myths and Legends'' (Jon Stewart, 1996) available at  [http://hegel.net/en/stewart1996.htm hegel.net] and at [http://www.marxists.org/reference/subject/philosophy/works/us/stewart.htm marxists.org], which will be want to reviewed by neophytes of Hegel literature.

== Essential Reading ==

=== Primary Sources ===

{{expand section|additional primary sources worthy of neophyte|date=July 2020}}
{| class=&quot;wikitable&quot; style=&quot;width:100%; margin:auto;&quot;
|-
! scope=&quot;col&quot; | {{abbr|No.|Number}}
! scope=&quot;col&quot; | Title
! scope=&quot;col&quot; | Author
! scope=&quot;col&quot; | Date
! scope=&quot;col&quot; | External URL
{{Book list|book_number=1|title=[[Encyclopaedia of the Philosophical Sciences]]|alt_title=Introduction|author=[[Georg Wilhelm Friedrich Hegel]]|short_summary=Overview of Hegel's main philosophical aims, view of his relation to other philosophical figures.|publish_date=1830|aux1=[https://www.marxists.org/reference/archive/hegel/works/sl/slintro.htm marxists.org]}}
{{Book list|book_number=2|title=[[The Spirit of Christianity and its Fate]]|author=[[Georg Wilhelm Friedrich Hegel]]|aux1=[https://www.marxists.org/reference/archive/hegel/works/fate/index.htm marxists.org]|publish_date=1798|short_summary=Hegel's early theological application of his philosophical principles.}}

|}
=== Secondary Sources ===
{{expand section|secondary sources worthy of neophyte|date=July 2020}}
===== Commentaries =====
==Biography==

[[Image:G.W.F. Hegel (by Sichling, after Sebbers).jpg|right|thumb|Georg Wilhelm Friedrich Hegel]]

Please consider the following articles:

* The [[w:Main_Page|Wikipedia]] entry for [[w:Georg_Wilhelm_Friedrich_Hegel|Hegel]].
* [http://www.marxists.org/reference/archive/hegel/help/hegelbio.htm Hegel's biography at marxists.org].
* [http://hegel.net/en/hegel.htm Hegel's biography at hegel.net].

==The ''Phenomenology of Spirit''==

The ''Phenomenology of Spirit'' (also translated as the ''Phenomenology of Mind'') is widely acknowledged as one of Hegel's most significant contributions to philosophy (perhaps his most important work). It is a very interesting and complex book. The 1911 [[w:Encyclopædia Britannica|Encyclopædia Britannica]] article on Hegel included a heroic attempt to summarize the ''Phenomenology'' which gives an idea of the far reaching implications of the work, from metaphysics to ethics and political philosophy:
* http://hegel.net/en/eb1911.htm#31
* [[Wikisource: 1911 Encyclopædia Britannica/Hegel, Georg Wilhelm Friedrich]]

* [[Hegel's Phenomenological Method|Hegel's Phenomenological Method]]

==Bibliographies==

* [http://www.sussex.ac.uk/Users/sefd0 Andrew Chitty] ([http://www.sussex.ac.uk University of Sussex, England]) has written a huge [http://www.sussex.ac.uk/Users/sefd0/bib/hegel.htm Hegel bibliography].
* [[w:Georg_Wilhelm_Friedrich_Hegel#Major_works|Wikipedia entry on Hegel]].
[[Category:Hegel]]
[[Category:Philosophy]]

==Active participants==

* [[User:Alex_beta|Alex beta]]</text>
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    <title>Trigonometry/Functions</title>
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      <comment>Sine theta can be equal to 1 at any point (n*pi)/2, where n is a whole number, and the same applies to cosine</comment>
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* See also '''[[Trigonometry/Polar]]''' for an approach that is useful for -&amp;infin;&lt;&amp;theta;&lt;&amp;infin;
==Fundamental trigonometry functions==

There are six trigonometric functions in Trigonometry: sine, cosine, tangent, cotangent, secant, and cosecant. 

:[[Image:TrigTriangle.svg|200px]]

==Sine==

* Sine θ is the length of the leg opposite θ over the length of the hypotenuse: &lt;math&gt;\sin\theta=\frac{opp}{hyp}&lt;/math&gt; 

==Cosine==
 
* Cosine θ is the length of the leg adjacent to θ over the hypotenuse: &lt;math&gt;\cos\theta=\frac{adj}{hyp}&lt;/math&gt;

==Tangent==

* Tangent of θ is the length of the leg on the opposite side of the triangle from the angle θ over the length of the leg of the triangle adjacent to the angle θ: &lt;math&gt;\tan\theta=\frac{opp}{adj}&lt;/math&gt; 

These three can be memorized by use of the name of the princess &quot;Soh Cah Toa,&quot; meaning:
* &quot;'''s'''ine-'''o'''pposite-'''h'''ypotenuse 
* '''c'''osine-'''a'''djacent-'''h'''ypotenuse 
* '''t'''angent-'''o'''pposite-'''a'''djacent&quot;.

The remaining ratios are reciprocals of the previous ratios: 

==Cotangent== 
* Cotangent θ is the reciprocal of tangent θ: &lt;math&gt;\cot\theta=\frac{adj}{opp}&lt;/math&gt;

==Secant==
* Secant θ is the reciprocal of cosine θ: &lt;math&gt;\sec\theta=\frac{hyp}{adj}&lt;/math&gt; 

==Cosecant==

* Cosecant θ is the reciprocal of sine θ: &lt;math&gt;\csc\theta=\frac{hyp}{opp}&lt;/math&gt;
&lt;br&gt;

==Other considerations==
*Since the hypotenuse of a right triangle is always the longest side, &lt;math&gt;opp &lt; hyp\,&lt;/math&gt; and &lt;math&gt;adj &lt; hyp\,&lt;/math&gt;
*If we divide both sides of each of these inequalities by the positive number &lt;math&gt;hyp\,&lt;/math&gt;, we get &lt;math&gt;\frac{opp}{hyp} &lt; \frac{hyp}{hyp}\,&lt;/math&gt; and &lt;math&gt;\frac{adj}{hyp} &lt; \frac{hyp}{hyp}\,&lt;/math&gt; &lt;br&gt;or &lt;math&gt;\sin \theta \leq 1\,&lt;/math&gt; and &lt;math&gt;\cos \theta \leq 1\,&lt;/math&gt;

==Table==
===Angle values===
{| class=&quot;wikitable&quot;
|-
! θ !! radians !! sinθ !! cosθ !! tanθ !! cotθ !! secθ !! cosecθ
|-
| 0° || 0 || 0 || 1 || 0 || undefined || 1 || undefined
|-
| 30° || π/6 || 1/2 ||     ||  ||  ||  || 2
|-
| 45° || π/4 ||     ||     || 1 || 1 ||  || 
|-
| 60° || π/3 ||     || 1/2 ||  ||  || 2 ||
|-
| 90° || π/2 || 1 || 0 || undefined || 0 || undefined || 1
|-
| 180° || π || 0 || -1 || 0 || undefined || -1 || undefined
|-
| 270° || 3π/2 || -1 || 0 || undefined || 0 || undefined || -1
|-
| 360° || 2π || 0 || 1 || 0 || undefined || 1 || undefined
|}

==Quiz==
*[[Trigonometry/Functions/Quiz|Trigonometry/Functions/Quiz]]

==Other resources==
*'''Reading''': [[w:Trigonometric_Functions]] (Wikipedia)
*'''Videos''':
:[http://www.youtube.com/watch?v=F21S9Wpi0y8&amp;feature=youtube_gdat Basic Trigonometry] (Youtube)
:[http://www.youtube.com/watch?v=QS4r_mqs-rY&amp;feature=youtube_gdata Basic Trigonometry II] (Youtube)
*[[Trigonometry/Functions/Flash cards]]

[[Category:Trigonometric functions]]</text>
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    <title>Home economics/How to clean a toilet</title>
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      <text bytes="10285" xml:space="preserve">All you need to do to clean up this page is read the page but replace toilet with page{{cleanup|There are some broken reference tags.}}

{{TOCright}}

[[Image:Toilet cleaning supplies.jpg|125px|right|Toilet cleaning supplies]]

[[Image:Disinfecting wipes.jpg|125px|right|Disinfecting wipes]]

There are two main parts to cleaning a [[w:toilet|toilet]]. The first is cleaning the outside of the toilet. The second is cleaning within the toilet bowl.(113) &lt;ref name=&quot;Smallin&quot;&gt;Smallin, Donna, ''Cleaning: plain &amp; simple,'' 2006, Storey Publishing, North Adams, MA, {{ISBN|978-1-58017-607-1}}; {{ISBN|1-58017-607-0}} (pb:  alk. paper)&lt;/ref&gt;

Caution needs to be used. If you get cleaning solution on your skin, wash it off with soap and water. If you get it in your eyes, rinse your eyes with water for twenty minutes.&lt;ref name=&quot;Rockline Industries&quot;&gt;Rockline Industries, ''Good &amp; Clean Disinfecting Wipes Lemon Scented Directions,'' Rockline Industries, Sheboygan, WI&lt;/ref&gt; Some people also worry about breathing in air born particles of cleaning chemicals.(24)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; They use cleaning wipes instead of spray.(118)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;&gt;Woodburn, Kim; MacKenzie, Aggie, ''How Clean Is Your House?'' Dutton, Penguin Group, New York, NY, {{ISBN|0-525-94857-0}}&lt;/ref&gt;

Toilet cleaning follows the general principals of cleaning any hard surface. Sweep the loose dirt and debris away. Soak the stuck dirt with cleaning solution. Wipe it and the solution away with a dry towel.(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;&gt;Aslett, Don; Phillips, Sandra, ''Don Aslett's Professional Cleaning System,'' 2005, InHomevations, Whittier, CA, {{ISBN|1-880759-76-4}}&lt;/ref&gt;

When cleaning the outside, work from the top to the floor.(61)&lt;ref name=&quot;Dellutri&quot;&gt;Dellutri, Laura, ''Speed Cleaning 101,'' Des Moines, Iowa, Meredith Books, 2005, {{ISBN|0-696-22414-3}}&lt;/ref&gt; When cleaning the toilet bowl, be careful not to splash.(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; 

Tools needed include rubber gloves(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;, disinfecting wipes, paper towels, plunger, scrubber, and toilet bowl cleaner to dissolve lime and calcium deposits.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 

==Outside==
===Step one===
{|
|
[[Image:Wipe down.jpg|125px|right|Wipe down]]
Put on rubber gloves. Pull a disinfecting wipe from the center of the roll up through the hole in the top of the container. To get one wipe to tear off from the next, pull the first wipe with one hand and stop the next wipe with the other hand. Close the little door on the top of the container each time to keep the wipes from drying out.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt; 
|}
===Step two===
{|
|
[[Image:Toss towelette.jpg|125px|right|Toss towelette]]
Rub part of the outside of the toilet with the wipe. This gets it wet with the cleaning solution in the wipe. Some of the dirt is picked up by the wipe. The rest starts floating in the cleaning solution. Get as much filth as possible off of that part of the toilet and on to the wipe. Throw the dirty wipe into a waste basket lined with a plastic bag.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;&lt;/ref&gt;(118 &amp; 119)&lt;/ref&gt;&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 
|}
===Step three===
{|
|
[[Image:Wipe up.jpg|125px|right|Wipe up]]
Tear an absorbent paper towel from its roll. Rub it on the area that was just wet by the wipe. The cleaning solution will be pulled off the surface and into the paper towel. The dirt floating in the cleaning solution will also. Dirt not floating but wet will be wiped off the area as well. Throw the dirty paper towel into the waste basket.(92 &amp; 110)&lt;ref name=&quot;Dellutri&quot;/&gt;(23)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;  
|}
===Step four===
{|
|
[[Image:Toss towel.jpg|125px|right|Toss towel]]
Repeat the process. Clean the top of the tank then the sides of the tank. Clean the bowl rim: first around the hinges. After cleaning the entire rim, do both sides of the lid and both sides of the seat. You might want to rinse the top of the seat with a wet paper towel and dry it with another paper towel. Next comes the back, sides and front of the toilet between the seat and the floor. If there is trash on the floor around the toilet, throw it away.(92)&lt;ref name=&quot;Dellutri&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Within== 
===Step five=== 
{|
|
[[Image:Safety measures.jpg|125px|right|Safety measures]]
Put on protective clothing, goggles, and rubber gloves. Do not get toilet bowl cleaner on clothing, skin or in eyes. It burns the skin, causes irreversible eye damage, and is very harmful if swallowed. If it gets in eyes, rinse eyes gently with water for twenty minutes keeping them open, removing any contact lenses after the first five minutes. If it gets on skin, remove contaminated clothing and rinse skin for twenty minutes. If any is swallowed, sip a glass of water if able to swallow. In all cases call a poison control center or a doctor. Wash contaminated clothing before wearing again.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
===Step six=== 
{|
|
[[Image:Water brushing.jpg|125px|right|Water brushing]]
With the toilet seat up, flush the toilet. Use the toilet brush with just the water to brush away any loose filth. Be cautious. Brush down to prevent liquid from being flicked at you by the bristles. Flush the toilet to rinse the bowl and the brush. Return the brush to its caddy. A looped toilet brush is able to lower the water level just a little.&lt;ref name=&quot;Reckitt Benckiser Inc&quot;&gt;Reckitt Benckiser Inc., ''Spring Waterfall Scent Lysol Brand Cling Gel Toilet Bowl Cleaner Directions,'' 2005, Reckitt Benckiser Inc., Parsippany, NJ&lt;/ref&gt; 
|}
===Step seven=== 
{|
|
[[Image:Lowering water level.jpg|125px|right|Lowering water level]]
Position the plunger so it covers the drain in the toilet bowl. Push downward on the plunger with both hands. Stop pushing and let the plunger fill with water again. Push downward on the plunger again so it forces the water within it down the drain. Repeat the push and release action several times, slowly enough for no splashing and fast enough to pump the water down the drain. The water would dilute the cleaning solution and might splash, so the water level needs to be as low as possible for safer more effective scrubbing.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;&gt;3M, ''Scotch-Brite Disposable Toilet Scrubbers Starter Pack Directions,'' 2006, 3M, St. Paul, MN&lt;/ref&gt;
|}
===Step eight===
{|
|
[[Image:Pouring cleaner.jpg|125px|right|Pouring cleaner]]
[[Image:Scrubbing the toilet.jpg|125px|right|Scrubbing the toilet]]
Open the bottle of toilet bowl cleaner being careful not to squeeze the bottle while doing so. With the toilet seat still up, slowly pour the toilet bowl cleaner from the bottle and into the toilet bowl. Carefully pour it around on the inner sides and no higher than under the rim. Close the cap and wipe the bottle with a paper towel if needed. Use a toilet scrubber or swab to brush the toilet bowl cleaner on to all the inside surfaces of the toilet bowl including under the rim. Wait a minute to allow the solution to soak into the dirt stuck on the surface. Scrub away the calcium deposits and dirt, or if it wasn't that dirty use a toilet bowl swab. Flush the toilet, rinse the scrubber or swab in the clean water and put it into its bucket. Wring the swab first. If the toilet bowl is still dirty wash it again with a bottle of stronger toilet bowl cleaner, if not already using the lime and calcium and rust removing kind.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(43, 110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;&gt;The Fuller Brush Company, ''Toilet Bowl Swab Directions,'' The Fuller Brush Company, Great Bend, KS, p. 18,29&lt;/ref&gt;&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; p. 113&lt;/ref&gt;&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Conclusion==
{|
|
[[Image:Toilet scrubber &amp; pads.jpg|125px|right|Toilet scrubber &amp; pads]]
[[Image:Toilet bowl swab.jpg|125px|right|Toilet bowl swab]]
If cleaner gets in eyes or on skin, rinse for twenty minutes and call a doctor. Read the labels of cleaning products.(35)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;&lt;ref name=&quot;Johnson &amp; Son&quot;&gt;S.C. Johnson &amp; Son, Inc., ''Scrubbing Bubbles Bathroom Cleaner Directions,'' 2006, S.C. Johnson &amp; Son, Inc., Racine, WI&lt;/ref&gt; 

The disinfecting wipe is more convenient than spray. Spray might miss the toilet and go on to the wall. It might run down the toilet and form a puddle on the floor. Pouring bathroom cleaner in a bucket and using it to clean the sponge or washcloth used on the toilet is more work than throwing a wipe into the trash.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;(112)&lt;ref name=&quot;Smallin&quot;/&gt;

Spraying the toilet bowl is awkward. Toilet bowl cleaner comes in a bottle designed to pour the cleaner around in the toilet bowl to cover the inside. There are scrub pads with short and with long handles. Some long handled scrub pads are small enough to scrub the drain. Disposable scrub pads are supposed to be thrown into the trash. A toilet brush can be used to scrub with by using a small circular motion. To use less cleaner the cleaner can be put in the bowl with swab or a folded paper towel.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;/&gt;(16, 27)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;

Wear protective clothing, safety goggles and rubber gloves.&lt;ref name=&quot;HomeCare Labs&quot;&gt;HomeCare Labs, ''The Works Disinfectant Toilet Bowl Cleaner directions,'' 2007, HomeCare Labs, Lawrenceville, GA&lt;/ref&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}

==References==

&lt;div class=&quot;references-small&quot;&gt;
&lt;references/&gt;
&lt;/div&gt;


[[Category:Home economics]]
[[Category:Tutorials]]
[[Category:Howtos]]</text>
      <sha1>0arxai8jhfcfnzdxr7wpqlqunm3g22r</sha1>
    </revision>
    <revision>
      <id>2624944</id>
      <parentid>2624937</parentid>
      <timestamp>2024-05-03T07:41:26Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>Reverted 1 edit by [[Special:Contributions/204.219.240.33|204.219.240.33]] ([[User talk:204.219.240.33|talk]]) (TwinkleGlobal)</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10199" xml:space="preserve">{{cleanup|There are some broken reference tags.}}

{{TOCright}}

[[Image:Toilet cleaning supplies.jpg|125px|right|Toilet cleaning supplies]]

[[Image:Disinfecting wipes.jpg|125px|right|Disinfecting wipes]]

There are two main parts to cleaning a [[w:toilet|toilet]]. The first is cleaning the outside of the toilet. The second is cleaning within the toilet bowl.(113) &lt;ref name=&quot;Smallin&quot;&gt;Smallin, Donna, ''Cleaning: plain &amp; simple,'' 2006, Storey Publishing, North Adams, MA, {{ISBN|978-1-58017-607-1}}; {{ISBN|1-58017-607-0}} (pb:  alk. paper)&lt;/ref&gt;

Caution needs to be used. If you get cleaning solution on your skin, wash it off with soap and water. If you get it in your eyes, rinse your eyes with water for twenty minutes.&lt;ref name=&quot;Rockline Industries&quot;&gt;Rockline Industries, ''Good &amp; Clean Disinfecting Wipes Lemon Scented Directions,'' Rockline Industries, Sheboygan, WI&lt;/ref&gt; Some people also worry about breathing in air born particles of cleaning chemicals.(24)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; They use cleaning wipes instead of spray.(118)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;&gt;Woodburn, Kim; MacKenzie, Aggie, ''How Clean Is Your House?'' Dutton, Penguin Group, New York, NY, {{ISBN|0-525-94857-0}}&lt;/ref&gt;

Toilet cleaning follows the general principals of cleaning any hard surface. Sweep the loose dirt and debris away. Soak the stuck dirt with cleaning solution. Wipe it and the solution away with a dry towel.(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;&gt;Aslett, Don; Phillips, Sandra, ''Don Aslett's Professional Cleaning System,'' 2005, InHomevations, Whittier, CA, {{ISBN|1-880759-76-4}}&lt;/ref&gt;

When cleaning the outside, work from the top to the floor.(61)&lt;ref name=&quot;Dellutri&quot;&gt;Dellutri, Laura, ''Speed Cleaning 101,'' Des Moines, Iowa, Meredith Books, 2005, {{ISBN|0-696-22414-3}}&lt;/ref&gt; When cleaning the toilet bowl, be careful not to splash.(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; 

Tools needed include rubber gloves(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;, disinfecting wipes, paper towels, plunger, scrubber, and toilet bowl cleaner to dissolve lime and calcium deposits.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 

==Outside==
===Step one===
{|
|
[[Image:Wipe down.jpg|125px|right|Wipe down]]
Put on rubber gloves. Pull a disinfecting wipe from the center of the roll up through the hole in the top of the container. To get one wipe to tear off from the next, pull the first wipe with one hand and stop the next wipe with the other hand. Close the little door on the top of the container each time to keep the wipes from drying out.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt; 
|}
===Step two===
{|
|
[[Image:Toss towelette.jpg|125px|right|Toss towelette]]
Rub part of the outside of the toilet with the wipe. This gets it wet with the cleaning solution in the wipe. Some of the dirt is picked up by the wipe. The rest starts floating in the cleaning solution. Get as much filth as possible off of that part of the toilet and on to the wipe. Throw the dirty wipe into a waste basket lined with a plastic bag.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;&lt;/ref&gt;(118 &amp; 119)&lt;/ref&gt;&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 
|}
===Step three===
{|
|
[[Image:Wipe up.jpg|125px|right|Wipe up]]
Tear an absorbent paper towel from its roll. Rub it on the area that was just wet by the wipe. The cleaning solution will be pulled off the surface and into the paper towel. The dirt floating in the cleaning solution will also. Dirt not floating but wet will be wiped off the area as well. Throw the dirty paper towel into the waste basket.(92 &amp; 110)&lt;ref name=&quot;Dellutri&quot;/&gt;(23)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;  
|}
===Step four===
{|
|
[[Image:Toss towel.jpg|125px|right|Toss towel]]
Repeat the process. Clean the top of the tank then the sides of the tank. Clean the bowl rim: first around the hinges. After cleaning the entire rim, do both sides of the lid and both sides of the seat. You might want to rinse the top of the seat with a wet paper towel and dry it with another paper towel. Next comes the back, sides and front of the toilet between the seat and the floor. If there is trash on the floor around the toilet, throw it away.(92)&lt;ref name=&quot;Dellutri&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Within== 
===Step five=== 
{|
|
[[Image:Safety measures.jpg|125px|right|Safety measures]]
Put on protective clothing, goggles, and rubber gloves. Do not get toilet bowl cleaner on clothing, skin or in eyes. It burns the skin, causes irreversible eye damage, and is very harmful if swallowed. If it gets in eyes, rinse eyes gently with water for twenty minutes keeping them open, removing any contact lenses after the first five minutes. If it gets on skin, remove contaminated clothing and rinse skin for twenty minutes. If any is swallowed, sip a glass of water if able to swallow. In all cases call a poison control center or a doctor. Wash contaminated clothing before wearing again.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
===Step six=== 
{|
|
[[Image:Water brushing.jpg|125px|right|Water brushing]]
With the toilet seat up, flush the toilet. Use the toilet brush with just the water to brush away any loose filth. Be cautious. Brush down to prevent liquid from being flicked at you by the bristles. Flush the toilet to rinse the bowl and the brush. Return the brush to its caddy. A looped toilet brush is able to lower the water level just a little.&lt;ref name=&quot;Reckitt Benckiser Inc&quot;&gt;Reckitt Benckiser Inc., ''Spring Waterfall Scent Lysol Brand Cling Gel Toilet Bowl Cleaner Directions,'' 2005, Reckitt Benckiser Inc., Parsippany, NJ&lt;/ref&gt; 
|}
===Step seven=== 
{|
|
[[Image:Lowering water level.jpg|125px|right|Lowering water level]]
Position the plunger so it covers the drain in the toilet bowl. Push downward on the plunger with both hands. Stop pushing and let the plunger fill with water again. Push downward on the plunger again so it forces the water within it down the drain. Repeat the push and release action several times, slowly enough for no splashing and fast enough to pump the water down the drain. The water would dilute the cleaning solution and might splash, so the water level needs to be as low as possible for safer more effective scrubbing.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;&gt;3M, ''Scotch-Brite Disposable Toilet Scrubbers Starter Pack Directions,'' 2006, 3M, St. Paul, MN&lt;/ref&gt;
|}
===Step eight===
{|
|
[[Image:Pouring cleaner.jpg|125px|right|Pouring cleaner]]
[[Image:Scrubbing the toilet.jpg|125px|right|Scrubbing the toilet]]
Open the bottle of toilet bowl cleaner being careful not to squeeze the bottle while doing so. With the toilet seat still up, slowly pour the toilet bowl cleaner from the bottle and into the toilet bowl. Carefully pour it around on the inner sides and no higher than under the rim. Close the cap and wipe the bottle with a paper towel if needed. Use a toilet scrubber or swab to brush the toilet bowl cleaner on to all the inside surfaces of the toilet bowl including under the rim. Wait a minute to allow the solution to soak into the dirt stuck on the surface. Scrub away the calcium deposits and dirt, or if it wasn't that dirty use a toilet bowl swab. Flush the toilet, rinse the scrubber or swab in the clean water and put it into its bucket. Wring the swab first. If the toilet bowl is still dirty wash it again with a bottle of stronger toilet bowl cleaner, if not already using the lime and calcium and rust removing kind.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(43, 110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;&gt;The Fuller Brush Company, ''Toilet Bowl Swab Directions,'' The Fuller Brush Company, Great Bend, KS, p. 18,29&lt;/ref&gt;&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; p. 113&lt;/ref&gt;&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Conclusion==
{|
|
[[Image:Toilet scrubber &amp; pads.jpg|125px|right|Toilet scrubber &amp; pads]]
[[Image:Toilet bowl swab.jpg|125px|right|Toilet bowl swab]]
If cleaner gets in eyes or on skin, rinse for twenty minutes and call a doctor. Read the labels of cleaning products.(35)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;&lt;ref name=&quot;Johnson &amp; Son&quot;&gt;S.C. Johnson &amp; Son, Inc., ''Scrubbing Bubbles Bathroom Cleaner Directions,'' 2006, S.C. Johnson &amp; Son, Inc., Racine, WI&lt;/ref&gt; 

The disinfecting wipe is more convenient than spray. Spray might miss the toilet and go on to the wall. It might run down the toilet and form a puddle on the floor. Pouring bathroom cleaner in a bucket and using it to clean the sponge or washcloth used on the toilet is more work than throwing a wipe into the trash.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;(112)&lt;ref name=&quot;Smallin&quot;/&gt;

Spraying the toilet bowl is awkward. Toilet bowl cleaner comes in a bottle designed to pour the cleaner around in the toilet bowl to cover the inside. There are scrub pads with short and with long handles. Some long handled scrub pads are small enough to scrub the drain. Disposable scrub pads are supposed to be thrown into the trash. A toilet brush can be used to scrub with by using a small circular motion. To use less cleaner the cleaner can be put in the bowl with swab or a folded paper towel.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;/&gt;(16, 27)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;

Wear protective clothing, safety goggles and rubber gloves.&lt;ref name=&quot;HomeCare Labs&quot;&gt;HomeCare Labs, ''The Works Disinfectant Toilet Bowl Cleaner directions,'' 2007, HomeCare Labs, Lawrenceville, GA&lt;/ref&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}

==References==

&lt;div class=&quot;references-small&quot;&gt;
&lt;references/&gt;
&lt;/div&gt;


[[Category:Home economics]]
[[Category:Tutorials]]
[[Category:Howtos]]</text>
      <sha1>o88t4vgz04p02vr64q9lcxgfz53po5c</sha1>
    </revision>
    <revision>
      <id>2624953</id>
      <parentid>2624944</parentid>
      <timestamp>2024-05-03T08:07:14Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>added [[Category:Toilets]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10220" xml:space="preserve">{{cleanup|There are some broken reference tags.}}

{{TOCright}}

[[Image:Toilet cleaning supplies.jpg|125px|right|Toilet cleaning supplies]]

[[Image:Disinfecting wipes.jpg|125px|right|Disinfecting wipes]]

There are two main parts to cleaning a [[w:toilet|toilet]]. The first is cleaning the outside of the toilet. The second is cleaning within the toilet bowl.(113) &lt;ref name=&quot;Smallin&quot;&gt;Smallin, Donna, ''Cleaning: plain &amp; simple,'' 2006, Storey Publishing, North Adams, MA, {{ISBN|978-1-58017-607-1}}; {{ISBN|1-58017-607-0}} (pb:  alk. paper)&lt;/ref&gt;

Caution needs to be used. If you get cleaning solution on your skin, wash it off with soap and water. If you get it in your eyes, rinse your eyes with water for twenty minutes.&lt;ref name=&quot;Rockline Industries&quot;&gt;Rockline Industries, ''Good &amp; Clean Disinfecting Wipes Lemon Scented Directions,'' Rockline Industries, Sheboygan, WI&lt;/ref&gt; Some people also worry about breathing in air born particles of cleaning chemicals.(24)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; They use cleaning wipes instead of spray.(118)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;&gt;Woodburn, Kim; MacKenzie, Aggie, ''How Clean Is Your House?'' Dutton, Penguin Group, New York, NY, {{ISBN|0-525-94857-0}}&lt;/ref&gt;

Toilet cleaning follows the general principals of cleaning any hard surface. Sweep the loose dirt and debris away. Soak the stuck dirt with cleaning solution. Wipe it and the solution away with a dry towel.(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;&gt;Aslett, Don; Phillips, Sandra, ''Don Aslett's Professional Cleaning System,'' 2005, InHomevations, Whittier, CA, {{ISBN|1-880759-76-4}}&lt;/ref&gt;

When cleaning the outside, work from the top to the floor.(61)&lt;ref name=&quot;Dellutri&quot;&gt;Dellutri, Laura, ''Speed Cleaning 101,'' Des Moines, Iowa, Meredith Books, 2005, {{ISBN|0-696-22414-3}}&lt;/ref&gt; When cleaning the toilet bowl, be careful not to splash.(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; 

Tools needed include rubber gloves(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;, disinfecting wipes, paper towels, plunger, scrubber, and toilet bowl cleaner to dissolve lime and calcium deposits.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 

==Outside==
===Step one===
{|
|
[[Image:Wipe down.jpg|125px|right|Wipe down]]
Put on rubber gloves. Pull a disinfecting wipe from the center of the roll up through the hole in the top of the container. To get one wipe to tear off from the next, pull the first wipe with one hand and stop the next wipe with the other hand. Close the little door on the top of the container each time to keep the wipes from drying out.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt; 
|}
===Step two===
{|
|
[[Image:Toss towelette.jpg|125px|right|Toss towelette]]
Rub part of the outside of the toilet with the wipe. This gets it wet with the cleaning solution in the wipe. Some of the dirt is picked up by the wipe. The rest starts floating in the cleaning solution. Get as much filth as possible off of that part of the toilet and on to the wipe. Throw the dirty wipe into a waste basket lined with a plastic bag.&lt;ref name=&quot;Rockline Industries&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;&lt;/ref&gt;(118 &amp; 119)&lt;/ref&gt;&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt; 
|}
===Step three===
{|
|
[[Image:Wipe up.jpg|125px|right|Wipe up]]
Tear an absorbent paper towel from its roll. Rub it on the area that was just wet by the wipe. The cleaning solution will be pulled off the surface and into the paper towel. The dirt floating in the cleaning solution will also. Dirt not floating but wet will be wiped off the area as well. Throw the dirty paper towel into the waste basket.(92 &amp; 110)&lt;ref name=&quot;Dellutri&quot;/&gt;(23)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;  
|}
===Step four===
{|
|
[[Image:Toss towel.jpg|125px|right|Toss towel]]
Repeat the process. Clean the top of the tank then the sides of the tank. Clean the bowl rim: first around the hinges. After cleaning the entire rim, do both sides of the lid and both sides of the seat. You might want to rinse the top of the seat with a wet paper towel and dry it with another paper towel. Next comes the back, sides and front of the toilet between the seat and the floor. If there is trash on the floor around the toilet, throw it away.(92)&lt;ref name=&quot;Dellutri&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(9)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Within== 
===Step five=== 
{|
|
[[Image:Safety measures.jpg|125px|right|Safety measures]]
Put on protective clothing, goggles, and rubber gloves. Do not get toilet bowl cleaner on clothing, skin or in eyes. It burns the skin, causes irreversible eye damage, and is very harmful if swallowed. If it gets in eyes, rinse eyes gently with water for twenty minutes keeping them open, removing any contact lenses after the first five minutes. If it gets on skin, remove contaminated clothing and rinse skin for twenty minutes. If any is swallowed, sip a glass of water if able to swallow. In all cases call a poison control center or a doctor. Wash contaminated clothing before wearing again.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}
===Step six=== 
{|
|
[[Image:Water brushing.jpg|125px|right|Water brushing]]
With the toilet seat up, flush the toilet. Use the toilet brush with just the water to brush away any loose filth. Be cautious. Brush down to prevent liquid from being flicked at you by the bristles. Flush the toilet to rinse the bowl and the brush. Return the brush to its caddy. A looped toilet brush is able to lower the water level just a little.&lt;ref name=&quot;Reckitt Benckiser Inc&quot;&gt;Reckitt Benckiser Inc., ''Spring Waterfall Scent Lysol Brand Cling Gel Toilet Bowl Cleaner Directions,'' 2005, Reckitt Benckiser Inc., Parsippany, NJ&lt;/ref&gt; 
|}
===Step seven=== 
{|
|
[[Image:Lowering water level.jpg|125px|right|Lowering water level]]
Position the plunger so it covers the drain in the toilet bowl. Push downward on the plunger with both hands. Stop pushing and let the plunger fill with water again. Push downward on the plunger again so it forces the water within it down the drain. Repeat the push and release action several times, slowly enough for no splashing and fast enough to pump the water down the drain. The water would dilute the cleaning solution and might splash, so the water level needs to be as low as possible for safer more effective scrubbing.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;&gt;3M, ''Scotch-Brite Disposable Toilet Scrubbers Starter Pack Directions,'' 2006, 3M, St. Paul, MN&lt;/ref&gt;
|}
===Step eight===
{|
|
[[Image:Pouring cleaner.jpg|125px|right|Pouring cleaner]]
[[Image:Scrubbing the toilet.jpg|125px|right|Scrubbing the toilet]]
Open the bottle of toilet bowl cleaner being careful not to squeeze the bottle while doing so. With the toilet seat still up, slowly pour the toilet bowl cleaner from the bottle and into the toilet bowl. Carefully pour it around on the inner sides and no higher than under the rim. Close the cap and wipe the bottle with a paper towel if needed. Use a toilet scrubber or swab to brush the toilet bowl cleaner on to all the inside surfaces of the toilet bowl including under the rim. Wait a minute to allow the solution to soak into the dirt stuck on the surface. Scrub away the calcium deposits and dirt, or if it wasn't that dirty use a toilet bowl swab. Flush the toilet, rinse the scrubber or swab in the clean water and put it into its bucket. Wring the swab first. If the toilet bowl is still dirty wash it again with a bottle of stronger toilet bowl cleaner, if not already using the lime and calcium and rust removing kind.&lt;ref name=&quot;HomeCare Labs&quot;/&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(43, 110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;&gt;The Fuller Brush Company, ''Toilet Bowl Swab Directions,'' The Fuller Brush Company, Great Bend, KS, p. 18,29&lt;/ref&gt;&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt; p. 113&lt;/ref&gt;&lt;ref name=&quot;Smallin&quot;/&gt;
|}
==Conclusion==
{|
|
[[Image:Toilet scrubber &amp; pads.jpg|125px|right|Toilet scrubber &amp; pads]]
[[Image:Toilet bowl swab.jpg|125px|right|Toilet bowl swab]]
If cleaner gets in eyes or on skin, rinse for twenty minutes and call a doctor. Read the labels of cleaning products.(35)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;&lt;ref name=&quot;Johnson &amp; Son&quot;&gt;S.C. Johnson &amp; Son, Inc., ''Scrubbing Bubbles Bathroom Cleaner Directions,'' 2006, S.C. Johnson &amp; Son, Inc., Racine, WI&lt;/ref&gt; 

The disinfecting wipe is more convenient than spray. Spray might miss the toilet and go on to the wall. It might run down the toilet and form a puddle on the floor. Pouring bathroom cleaner in a bucket and using it to clean the sponge or washcloth used on the toilet is more work than throwing a wipe into the trash.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(109)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;Rockline Industries&quot;/&gt;(112)&lt;ref name=&quot;Smallin&quot;/&gt;

Spraying the toilet bowl is awkward. Toilet bowl cleaner comes in a bottle designed to pour the cleaner around in the toilet bowl to cover the inside. There are scrub pads with short and with long handles. Some long handled scrub pads are small enough to scrub the drain. Disposable scrub pads are supposed to be thrown into the trash. A toilet brush can be used to scrub with by using a small circular motion. To use less cleaner the cleaner can be put in the bowl with swab or a folded paper towel.(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(110)&lt;ref name=&quot;Dellutri&quot;/&gt;&lt;ref name=&quot;3M DTS Directions&quot;/&gt;&lt;ref name=&quot;Reckitt Benckiser Inc&quot;/&gt;&lt;ref name=&quot;Fuller Brush&quot;/&gt;(16, 27)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(113)&lt;ref name=&quot;Smallin&quot;/&gt;

Wear protective clothing, safety goggles and rubber gloves.&lt;ref name=&quot;HomeCare Labs&quot;&gt;HomeCare Labs, ''The Works Disinfectant Toilet Bowl Cleaner directions,'' 2007, HomeCare Labs, Lawrenceville, GA&lt;/ref&gt;(118 &amp; 119)&lt;ref name=&quot;Woodburn &amp; MacKenzie&quot;/&gt;(35, 49)&lt;ref name=&quot;Dellutri&quot;/&gt;(29)&lt;ref name=&quot;Aslett &amp; Phillips&quot;/&gt;(19, 254)&lt;ref name=&quot;Smallin&quot;/&gt;
|}

==References==

&lt;div class=&quot;references-small&quot;&gt;
&lt;references/&gt;
&lt;/div&gt;


[[Category:Home economics]]
[[Category:Tutorials]]
[[Category:Howtos]]
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[[/2007-2018]]}}

== [[WikiJournal of Medicine/Medical gallery of Blausen Medical 2014]] ==

Back in 2014, you said some images were off-topic because they are mostly related to chemistry or plants and thus not suitable for WJM. Is it a suitable time to publish them in WJS? (Tagging {{u|Evolution and evolvability}}) [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 07:14, 9 December 2018 (UTC)

:{{re|OhanaUnited}} Interesting idea. In addition to the images omitted from the 2014 publication, there are a couple of non-med images uploaded since then, such as [[:File:Heat_Transfer.png]] (though they could have more non-uploaded ones). I have noticed though, that they have uploaded [https://commons.wikimedia.org/w/index.php?title=Special:Contributions/BruceBlaus&amp;offset=&amp;limit=500&amp;target=BruceBlaus &gt;500 new med images since 2015], so a selection of the best could also be viable WikiJMed publication. Perhaps we could contact them about both ideas? [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 10:46, 9 December 2018 (UTC)

::{{u|OhanaUnited}} and {{u|Evolution and evolvability}}, I agree these are indeed potential submissions, but although I'd like to help I'm currently very busy with other projects. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 21:28, 9 December 2018 (UTC)

:::No problem. I'll send an email to Blausen Medical with you and {{u|OhanaUnited}} cced in. Feel free to archive any emails, they'll just be so you have a record. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 22:53, 9 December 2018 (UTC)

:::: How did that medical gallery's peer review process go? I don't mind coordinating if it's not overly complex. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 00:36, 10 December 2018 (UTC)
==Hep E==
:Dr. Häggström, have done requested edit[https://en.wikiversity.org/w/index.php?title=Talk:WikiJournal_Preprints/Hepatitis_E&amp;diff=prev&amp;oldid=1973076], thank you--[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 21:14, 5 February 2019 (UTC)
:*Have finished ''second part'' as well, thank you--[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 00:55, 6 February 2019 (UTC)

== All WJM ms rejected ==

Is there any reason why all manuscripts in WJM are rejected? Or are you closing down the journal? All pages have this statement &quot;This article has been declined for publication by the WikiJournal of Medicine. It is archived below as a record.&quot; [[User:Chhandama|Chhandama]] ([[User talk:Chhandama|discuss]] • [[Special:Contributions/Chhandama|contribs]]) 09:07, 26 March 2019 (UTC)
:{{re|Chhandama}} apologies - that was my fault! I accidentally broke a formatting template! I've fixed it now. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 09:25, 26 March 2019 (UTC)

==Upcoming 2019 Affiliate-selected trustee position on the Board of Wikimedia Foundation==

Greetings {{ping|Mikael Häggström}} and Fellow Wikimedians,

My name is Gerald Shields, also known as user:Geraldshields11. I am asking for your top rank vote for me in the election for one of two open trustee positions on the Wikimedia Foundation Board of Trustees. 

I am asking for your vote to help support emerging communities and promote an inclusive education environment on all wiki projects. Also, I plan to promote various other issues as I mention in my statement and answers to various questions. I ask that you show your support for the issues that need to be address by voting for me as one of your preferences.

As of the 7 May 2019 list of questions, I have responded to all of the “Questions for all candidates - Questions for this individual candidate that do not apply to other candidates”. My answers give more details on why the affiliate should vote for me.

My candidacy information page is [[Affiliate-selected Board seats/2019/Nominations/Gerald Shields]] on Meta or can be found at [https://meta.wikimedia.org/wiki/Affiliate-selected_Board_seats/2019/Nominations/Gerald_Shields Gerald Shields candidate]. 
Thank you for your time, discussion, and consideration of my candidacy. I appreciate it. 

My best regards,

Gerald Shields

[[User:Geraldshields11|Geraldshields11]] ([[User talk:Geraldshields11|discuss]] • [[Special:Contributions/Geraldshields11|contribs]]) 12:36, 9 May 2019 (UTC)

:Thank you for your organization's fourth group vote during the 2019 trustee election. With your organizations', Florida's, Igbo's, and AfroCrowd's votes, I was able to score better than expected. [[User:Geraldshields11|Geraldshields11]] ([[User talk:Geraldshields11|discuss]] • [[Special:Contributions/Geraldshields11|contribs]]) 16:27, 18 June 2019 (UTC)

::I'm glad you were happy with the election outcome. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 22:56, 19 June 2019 (UTC)

== Broken image on main pge ==

Sorry, I didn't know whom to ping, but today's featured project section on the main page has a broken image.  Can you please ping the right person to fix it? [[User:Acagastya|Acagastya]] ([[User talk:Acagastya|discuss]] • [[Special:Contributions/Acagastya|contribs]]) 06:00, 3 May 2024 (UTC)</text>
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    <title>Perpendicular and parallel lines</title>
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      <text bytes="4669" xml:space="preserve"> line q has a slope of -6 according to the slopes of perpendicular lines postulate which lines are perpendicular to line q

as long as neither is vertical. Then a and c are the slopes of the two lines. The lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1.

The angle times the height of another angle equals the sum of one angle. The perpendicular force is equivalent to the base and also the height of the vertex/reflex angle(s). Perpendicular lines have negative reciprocal slopes. So perpendicular means a 90 degree angle at at least two sides of the symbol. It has to have a line sticking straight up above a horisontal line.

To construct the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).

    * Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P.
    * Step 2 (green): construct circles centered at A' and B', both having a radius greater than A'P. Let Q be the other point of intersection of these two circles.
    * Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal.

As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In Figure 3, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

    * One of the angles in the diagram is a right angle.
    * One of the orange-shaded angles is congruent to one of the green-shaded angles.As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

If lines a and b are parallel, any of the following conclusions leads to all of the others:

    * One of the angles in the diagram is a right angle.
    * One of the orange-shaded angles is congruent to one of the green-shaded angles.
    * Line 'c' is perpendicular to line 'a'.
    * Line 'c' is perpendicular to line 'b'.

    * Line 'c' is perpendicular to line 'a'.
    * Line 'c' is perpendicular to line 'b'.

Algebra

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. It is helpful to memorize the slogan &quot;to find the slope of the perpendicular line, flip the fraction and change the sign.&quot; Recall that any whole number a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.
 Calculus

First find the derivative of the function. This will be the slope (m) of any curve at a particular point (x, y). Then, as above, solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range: ⟂.


== Parallel Lines ==
 
Lines are parallel if they are always the same distance apart (called &quot;equidistant&quot;), and will never meet. Just remember:
Parallel lines always the same distance apart and never touch.

Parallel lines also point in the same direction.

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same.

These angles can be made into pairs of angles which have special names.

[[Category:Geometry]]</text>
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*'''May 2:''' The [[Living Wisely]] course instructor is now [[Living Wisely#Objectives|holding regular office hours]]. 
*'''March 15:''' [[User:Jtneill/Presentations/Using open wikis for teaching and learning|Using open wikis for teaching and learning]] presentation to the ASCILITE Learning Design Special Interest Group
*'''January 5:''' The [[Real Good Religion]] course is now welcoming students seeking adventure

'''2023'''
*'''December 25:''' The recently revised course on [[writing]] is now available
*'''May 23:''' Students can now follow [[The Wise Path]] on Wikiversity
*'''April 11''': The course [[Collective decision making]] is now available as part of the [[Coming Together]] curriculum
*'''January 1''': [[Socialism]] is calling for high quality essays&lt;noinclude&gt;
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{{tlf|editprotected}}
This page is under a decade-old full protection and is untouched since 2015.
Could it please be lowered to semi-protetion to allow for constructive edits by established users? 
[[User:Elominius|Elominius]] ([[User talk:Elominius|discuss]] • [[Special:Contributions/Elominius|contribs]]) 17:45, 8 February 2023 (UTC)
:&lt;s&gt;&lt;nowiki&gt;{{ping|Elominius}} Are you sure this page it protected?  The template says it's protected, but my attempt to unprotect it suggests that there is no protection.  {{ping|Dave Braunschweig}} Is there something I am not seeing here? --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:26, 12 February 2023 (UTC)&lt;/nowiki&gt;&lt;/s&gt;
::Duh! You wanted [[Help:User page]] unprotected, not [[Help:User talk:User page]].  I unprotected it.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:30, 12 February 2023 (UTC)
::: Thank you for changing the protection, Guy! Much appreciated. I have already made an edit. ☺ [[User:Elominius|Elominius]] ([[User talk:Elominius|discuss]] • [[Special:Contributions/Elominius|contribs]]) 07:52, 13 February 2023 (UTC)

== BRICS summit 2023 ==

Po [[User:Nelisa Sithonga|Nelisa Sithonga]] ([[User talk:Nelisa Sithonga|discuss]] • [[Special:Contributions/Nelisa Sithonga|contribs]]) 16:16, 28 August 2023 (UTC)

== Global Blood Glucose Monitoring Devices Market Share by 2035 ==

The Global [https://www.rootsanalysis.com/reports/diabetes-monitoring-devices-market.html Blood Glucose Monitoring Devices Market] Research Report added by Roots Analysis to its expanding repository is an all-inclusive document containing insightful data about the Blood Glucose Monitoring Devices market and its key elements. The report is formulated through extensive primary and secondary research and is curated with an intent to offer the readers and businesses a competitive edge over other players in the industry. The report sheds light on the minute details of the Blood Glucose Monitoring Devices industry pertaining to growth factors, opportunities and lucrative business prospects, regions showing promising growth, and forecast estimation till 2032. The report assesses the historical data and current scenario to offer accurate estimations of the Blood Glucose Monitoring Devices market in the coming years. The blood glucose monitoring devices market is estimated to be worth $16 billion in 2020 and is expected to grow at CAGR of 8% during the forecast period.

Furthermore, the report provides a comprehensive analysis of the factors that are likely to bolster or impede the growth of the market in the coming years. The report considers the COVID-19 pandemic that is currently unfolding as a key market influencer. The report provides a thorough estimation of the overall impact of the pandemic on the Blood Glucose Monitoring Devices market and its vital segments. The report also discusses the impact of the pandemic across different regions of the market. It also offers a current and future assessment of the impact of the pandemic on the Blood Glucose Monitoring Devices market.

'''Get a sample of the report @ https://www.rootsanalysis.com/reports/diabetes-monitoring-devices-market/request-sample.html''' [[User:Rseearchtrends1|Rseearchtrends1]] ([[User talk:Rseearchtrends1|discuss]] • [[Special:Contributions/Rseearchtrends1|contribs]]) 11:55, 3 May 2024 (UTC)</text>
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{{tlf|editprotected}}
This page is under a decade-old full protection and is untouched since 2015.
Could it please be lowered to semi-protetion to allow for constructive edits by established users? 
[[User:Elominius|Elominius]] ([[User talk:Elominius|discuss]] • [[Special:Contributions/Elominius|contribs]]) 17:45, 8 February 2023 (UTC)
:&lt;s&gt;&lt;nowiki&gt;{{ping|Elominius}} Are you sure this page it protected?  The template says it's protected, but my attempt to unprotect it suggests that there is no protection.  {{ping|Dave Braunschweig}} Is there something I am not seeing here? --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:26, 12 February 2023 (UTC)&lt;/nowiki&gt;&lt;/s&gt;
::Duh! You wanted [[Help:User page]] unprotected, not [[Help:User talk:User page]].  I unprotected it.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:30, 12 February 2023 (UTC)
::: Thank you for changing the protection, Guy! Much appreciated. I have already made an edit. ☺ [[User:Elominius|Elominius]] ([[User talk:Elominius|discuss]] • [[Special:Contributions/Elominius|contribs]]) 07:52, 13 February 2023 (UTC)

== BRICS summit 2023 ==

Po [[User:Nelisa Sithonga|Nelisa Sithonga]] ([[User talk:Nelisa Sithonga|discuss]] • [[Special:Contributions/Nelisa Sithonga|contribs]]) 16:16, 28 August 2023 (UTC)</text>
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__TOC__
{{MECR|1=http://www.youtube.com/watch?v=dIrqOnofoIU}}

==Overview==
[[file:Felicidade A very happy boy.jpg|250px|right]]In the early centuries of human civilisation, religiosity stated that the antecedents of happiness were immortality, wisdom and bliss; and as such, many believed that happiness was experienced only by the gods (McMahon, 2010). But in 380 BC Socrates (cited in McMahon) asks “What being is there who does not desire happiness?... How can we be happy?” With these questions, Socrates transforms the notion of happiness not only into something which mortal humans can experience, but he also suggests the importance of happiness by prompting the search for its causes. 

Happiness is something which people are said to strive for and pursue as often as possible (McMahon, 2010). Happiness can be spoke of in terms of life-long happiness, current situational happiness and contentment (see Griffin, 2007). It can be discussed terms of moods, emotions, attitudes and feelings (Griffin) which relate to different [[w:Cognition | cognitive]], [[w:Physiology | physiological]], and durational experiences. Happiness can be experienced when one feels fortunate, content, cheerful or glad (Griffin). Forethought can produce happiness when thinking about an upcoming holiday, as can positive feedback when one receives a great mark at school. Happiness can be something that is felt when needs have been fulfilled or goals have been reached, or in a moment of 'bliss' (Jacobsen, 2007). The goals for which people strive are different for each person, and a blissful moment for one may be anxiety producing for another; where one may see the end of a romantic relationship as an invigorating start to a new journey, and another may feel that their world has come to an end. 

This subjective nature of the term 'happiness' has prompted much research over many years. Most researchers seem to have agreed that 'subjective well-being' is a reasonable way to operationalise the concept of happiness (Diener &amp; Ryan, 2009; Griffin, 2007; Jacobsen, 2007; Natvig, Albrektsen, &amp; Qvarnstrom, 2003). Confirming the valid use of this term as an operational definition, Davern et al. deconstructed scales of subjective well-being, to see what they were actually measuring. The authors found six [[w:Affect (psychology) | affective]] descriptions contained in tests of subjective well-being which explained 64% of the variance of how satisfied one is with their life on a whole. These affects included which energised, happy, content, satisfied, pleased and negatively correlated stressed, which do indeed seem to be reasonable encompassing of the term 'happiness'. Subjective well-being is often measured by asking how happy one is with their life on a whole (Davern, Cummins &amp; Stokes, 2007) and will be the operational definition of happiness used in this chapter.

==Cognitive Models==

[[File:Happy pose I.JPG|250px|left]]
===Construal Model===
The construal model is a cognitive model of happiness that involves explaining the experience of happiness in terms of how people interpret (or construe) an event (Lyubomirsky, 2001). Lyubomirsky explains that the subjectivity involved in this process explains why some individuals can continually interpret events in a way which undermines their happiness, while others construe events in a manner which maintains or promotes their happiness. Individuals can differ in their understandings by construing an event as either challenging or threatening; with a sense of humour or tragedy; by seeing it favourably or unfavourably; and by dwelling on the situation or avoiding thoughts of it. These are cognitive strategies which can either help or hinder the experience of happiness. Lyubomirsky and Tucker (1998) investigated such strategies and found five that happy people are adept at using adverse situations. These include ''I just enjoyed the event and didn't think about it too much''; ''I found something positive and productive about the event (e.g. thought about what was learned from it, looked at its positive aspects, thought about the accomplishment of having survived it)''; ''I looked at the event with a sense of humour''; ''I avoided dwelling on the event'' and ''I thought about how much better things are now''.

In their second study, Lyubomirsky and Tucker (1998) further investigated these cognitive strategies. Participants completed a 'good-feelings' questionnaire and were either classified as being generally happy or generally unhappy. They asked participants to describe an event which had made them happy, and one which had made them unhappy. Participants then gave the stories ratings on a 7 point likert scale (1 = very unhappy, 7 = very happy). These stories were then transcribed and randomly allocated to impartial judges who gave their own ratings on how happiness inducing the events were. This was done in an attempt to decipher an objective happiness rating of the events. The authors found that generally happy gave their happy memories higher ratings than did unhappy people, and unhappy people described their less fortunate events as more disheartening. Judges provided confirmation of the subjective nature of happiness when they failed to find any differences in [[w:Affect (psychology) | affect]] between the stories of happy and unhappy people. That is to say, even though happy people rated their positive life event more highly, the judges did not interpret these events to be more positive. The same effect was observed for unhappy stories, where the judges could not see a difference between events described by happy and unhappy people. If the situations themselves do not differ in happiness, then it must be some difference in how the event is perceived which provokes more elated feelings. This provides strong evidence in support of the construal model of happiness. 

In their third study, Lyubomirsky and Tucker (1998) found that happy people dealt with situations in a more adaptive way. Happy people were better able to make decisions which may have been slightly [[w:Aversives | aversive]] in the short term, but which provide substantial benefits in the long term. They found that unhappy people were more likely to distract themselves from a problem in the short-term, avoiding dealing with the issue, which can lead to the problem being a source of more ill-feeling down the track. This ability to deal with an issue head-on may well help sustain happiness in happy people in the long-term, yet another benefit of having a happy disposition. But is it peoples happiness which leads them to deal with issues head-on, or is it dealing with issues head-on that leads to happiness? Earlier it was explained that events rated similarly by impartial judges were seen to be the source of significantly more happiness for already happy individuals suggesting that cognitions, not the event itself, causes positive affect.

===Self Attention Model===
Self attention is another cognitive process said to moderate feelings of happiness. Self attention can lead to either an increase or decrease in subjective well-being depending on which process is used: self-reflection or self-rumination. Self-reflection is a process related to honest curiosity about the self, where one is interested in knowing more about their inner thought processes, attitudes, emotions and vales; and can lead to wisdom (Morin, 2002). On the other hand, self-rumination is self-thought with undertones of anxiety, self doubt and lack of self-worth (Morin), which leads to sadness (Trapnell &amp; Campbell, 1999) Whereas the person who self-reflects is curious to learn more, a person who self-ruminates is self-criticizing. Trapnell and Campbell tested a concept called the ''Self-absorption paradox'', where those who are self-reflective are more self-knowledgeable and wise, but are also prone to pay attention to negative aspects of the self, which can lead to sadness. In their investigation, Trapnell and Campbell found their self-attentive participants were either sadder ''or'' more wise, not both. They found that greater self-reflection led to greater happiness, but not if the person was prone to self-rumination. If a person was highly self-reflective and highly self-ruminative they experienced more negative affect than most. This study suggests that self-attention does not automatically lead to sadness as was suggested by the Self-absorption paradox. Instead, if self-attention is the result of a genuine curiosity to gain further knowledge about the self, and if a person can disengage these reflective processes when it is leading to negative feedback, then self-attention can lead to increased happiness (Trapnell &amp; Campbell). This result has been replicated in other studies (e.g. Elliott &amp; Coker, 2008). It is possible that this effect is observed because a predominately self-reflective person sees their shortcomings as constructive feedback, leading to more positive behaviour in the future.

===Discrepancy Theory===
Another cognitive theory, know as Discrepancy Theory, suggests that affective experiences are the result of [http://dictionary.reference.com/browse/discrepancy discrepancies] between the actual self, the ideal self, and the ought self (McDaniel &amp; Grice, 2008). McDaniel and Grice explain that the ideal self is what a person strives to be, and the ought self is who the person thinks they ought to be as suggested by people around them. In discrepancy theory, higher discrepancy between the actual and ideal selves is said to predict depression, and discrepancy between actual and ought self is said to predict anxiety (McDaniel &amp; Grice). In their study, McDaniel and Grice found both actual-ideal and actual-ought discrepancies to be highly correlated (p ≥.90), as such they could not be said to represent different constructs. They found discrepancies in the actual-ideal/ought selves did predict variance in depression, self esteem and anxiety. Insomuch as depression is negatively correlated with happiness, it can be said that those with less actual-ideal/ought discrepancies will experience increased happiness. An extension of Discrepancy Theory, called Multiple Discrepancy Theory, suggests that happiness (or lack of) is experienced through perceived discrepancies in a) what someone has now, b) what they want in the future, c) what the best is that they've had in the past, and d) what they feel they deserve and what they need (Davern et al., 2007).
{{Hide in print|
===Cognitive Models: Quiz Question===
&lt;quiz display=simple&gt;
{ When does self-attention lead to happiness?
| type=&quot;()&quot; }
- Always
+ When it involves a genuine curiosity in ones self
- When one focuses on the negative aspects of the self
- Never
&lt;/quiz&gt;
}}
==Other Models==
===Personality===
Despite the understanding that environmental events which are generally fluid can influence the experience of happiness, measures of subjective well-being seem to be fairly stable across populations across time (Jacobsen, 2007). Happiness has been seen to vary by only 3.3% in 4 years in an Australian population (Cummins et al, cited in Davern et al., 2007). Davern and colleagues propose that evolution has predisposed humans to have a 'set point' for happiness, an average overall happiness of approximately 75%. Due to this stability, and the stability of personality over time, it is suggested that happiness may be primary attributed to the [[w:Big Five personality traits | big 5 personality traits]] (e.g. Hotard, McFatter, McWhirter, Stegall, 1989). Through [[w:Path analysis (statistics) | path models]] Davern and colleagues found that Core Affects of content, happy and excited, actually influence personality and subjective well-being, rather than personality being the causal factor.

===Evolution===
As far as the evolution of happiness goes, Nesse (2005) explains that just as negative emotions have evolved to help us avoid adverse circumstances, positive emotions have evolved to help us approach situations which hold opportunities. He explains explains the evolution of happiness in terms of very distant relatives of humans who are known as bacteria. Nesse explains that a bacterium continue in a direction if that direction is providing it with the same or better circumstances to what it was in before. If the direction the bacterium is travelling in provides worse circumstances, it will tumble into a randomly new direction. Nesse suggests that the emotions humans have today are actually just sophisticated variations of the behaviour of bacteria. He propses that as organisms evolved they required differentiated emotional responses to deal with ever increasing social, personal and reproductive choice.

==Happy ==
[[File:Happy Old Man.jpg|right|400px]]
===Buffering Effects===
Subjective well-has been investigated using Social Comparison theory. In this theory, ''upward comparison'' happens when people compare themselves to someone perceived to be more competent, and tends to lead to a decrease in self-esteem and well-being (Lyubomirsky &amp; Ross, 1997). ''Downward comparisons'' which are comparisons to people perceived to be less competent or able, can be reassuring and self-enhancing (Lyubomirsky &amp; Ross). This method of behiaviour is a source of great differences between individuals; where some rely heavily on social comparisons to form their self concept, and others rely on it very little (Lyubomirsky &amp; Ross). Lyubomirsky and Ross aimed to investigate how happy and unhappy people differ in methods and outcomes of social comparison. Prior to the task, participants were classified as either happy or unhappy individuals. Then after an anagram solving task, social comparison was induced by randomly reporting to participants whether a peer had performed better or worse than they. Participants completed measurements of mood, self-rated ability and self-confidence pre- and post-task. The authors found that after completing the task and being told that they had out-performed their peer, both happy and unhappy participants rated their ability as higher than they did pre-task. When participants were told that they were outperformed by a peer, those classified as happy still felt that their ability had increased post-task, but those will low happiness reported a decreased perception of their ability. A similar trend was observed in ratings of mood. When individuals were out-performed by a peer, 'happy' participants had no change in mood pre- and post-task, but those considered to be unhappy had a significantly lower mood post-task. Thus, having a happy disposition can act as a buffer to the negative effects of social comparison. Happy people seem to be better able to ignore or defend themselves against the negative impact of social comparisons.

In Lyubomirsky and Ross's second study (1997), they further investigated these effects. In one condition participants were given fake feedback that they had done well but a peer had done better; and in the second condition, the feedback suggested that the participant had performed averagely, but their peer had performed worse. They found that 'happy' individuals exhibited increased positive mood after being given a positive evaluation even though their peer had scored higher, and a slightly deflated mood being given a mediocre evaluation, even though their peer had scored lower. The opposite was true of unhappy individuals who displayed elevated mood when given mediocre feedback and doing better than their peer, and deflated mood after having done very well but being outperformed by their peer. Through social comparison, 'unhappy' people experienced ill feelings after having done well and good feelings after having done not-so-well, which is quite maladaptive behaviour. This result suggests that social comparison can be particularly detrimental to those who experience 'unhappy' dispositions. 

===Physical Health===
Veenhoven (2008) conducted a meta-analysis of longitudinal studies with the main aim of finding out if happiness produces better health, and if so, how this process occurs. Veenhoven found that in 53% of studies participants with higher levels of happiness were found to live longer, in 34% there was found to be no difference in live expectancy between happy and unhappy individuals, and in 13% of the studies, happy people were actually found to have a decreased life expectancy. In non-chronically ill populations happy participants were found to live for an average of 8.75 years longer than those who are unhappy. Veenhoven explains that happiness has never been found to ''cure'' disease, so instead describes ways in which happiness must be contributing to some kind of disease prevention. Veenhoven suggests that unhappiness places strain on the fight-or-flight response. This can cause long term lower immune system functioning and higher blood pressure, which can account for much of the variance in longevity between happy and unhappy people. Happiness also leads to healthier lifestyle behaviour, like exercise and healthy eating (Veenhoven).

===Work Success, Family and Aging===
Lyubomirsky, King and Diener (2005) compiled a meta-analytic study of 225 papers and found that happy people were more likely to be interviewed, received better ratings from supervisors and exhibited higher performance and productivity. Happy people were also found to be better able to cope with the demands in managerial positions, and received higher incomes (Lyubomirsky et al.). Other studies have also found this to be the case, but that the benefits of income on happiness taper off towards higher earning brackets (around $100,000 or more) (North, Holahan, Moos &amp; Cronkite, 2008). However, Garđarsdóttir, Dittmar, Aspinall (2009) found that the want for money has a negative correlation with happiness, possibly because those who dedicate more time to pursuing money have less time and to spend on other happiness inducing activities. In the North and colleagues article mentioned above, it was also found that happiness and family support are significantly correlated. North and colleagues scale of family support measured how much family members helped and backed each others decisions and how much family members were encouraged to openly and honestly express their feelings to one another. North and colleagues' results suggest that happiness can be increased if these attitudes of expressiveness and supportiveness are adopted, which will also foster feelings acceptance and belonging within the family. Despite some common stereotypes of elderly people, many studies have found that happiness does not decrease with age, and can actually increase (Ersner-Hershfield, Mikels,  Sullivan, &amp; Carstensen, 2008; Lacey, Smith, &amp; Ubel, 2006). This is said to be prompted by a greater appreciation of a person's mortality, making time more meaningful (Ersner et al.). In turn this prompts older people to focus on more emotionally meaningful goals (Ersner-Hershfeild et al.), an attitude that other researches (e.g. Tkach &amp; Lyubomirsky, 2006) have found to increase happiness.
{{Hide in print|
===Benefits of Happiness: Quiz Question===
&lt;quiz display=simple&gt;

{What is ''upward comparison''?
|type=&quot;()&quot;}
- When one thinks they are better than someone else
- When one strives to achieve their best
+ When one compares them self to someone who they think has more ability
&lt;/quiz&gt;
}}
==Increasing Happiness==
[[File:Happy boys.jpg|300px|left]]Happiness is said to be a function of living skills such as social competance and determination (Veenhoven, 2008). Tkach and Lyubomirsky (2006) aimed to elucidate such living skills by asking participants to give rate 66 happiness increasing attitudes on how often they were employed. Six of these attitudes appeared to be most used in producing and maintaining happiness, and could account for 52% of the total variance in happiness scores.  The first of these attitudes is called ''social affiliation'' which involves helping others and communicating with friends. The second is ''mental control'', an attitude inversely related to happiness, where suppression of emotional expression and a need for control over emotions seems to lead to unhappiness. Then there is ''instrumental goal pursuit'', where happiness can be increased by committing to personally meaningful goals. Next there is ''active leisure'', where a person makes a marked effort to participate in exercise and hobbies. Active leisure increases positive affect, especially compared ''passive leisure'' like sleeping and watching TV, which have been found to have no effect on well being. Then there is the rarely used but nonetheless beneficial involvement in religion. Religion is said to increase happiness because it provides people with a sense of being socially connected and often gives people a sense of purpose in their life. Finally, ''direct expression of happiness'' is correlated with increased experience of happiness, but the causal direction of this relationship is unclear (Tkach &amp; Lyubomirsky). It is possible that people who internally experience happiness are more likely to outwardly express it; but since the years of enlightenment, it has been known that outward expressions of happiness cause an increase of internal positive affect (Tkach &amp; Lyubomirsky). 

In their meta-analysis of 225 research papers on happiness, Lyubomirsky and colleages (2005) found several other actions and attitudes which contribute to a more happy disposition. These include activity and sociability, positive perceptions of ones self and of others, cooperation and likability, creativity and problem solving, coping and physical well-being, and prosocial behaviour (Lyubomirsky et al. 2005).  In 2008, Robinson and Martin analysed 34 years of data collected through the [[w:General Social Survey | General Social Survey]] in the United States. They found that people who socialise more with relatives and friends seem to be happier, as are people who go to church more often. Robinson and Martin also found that amount of hours spent at work did not significantly effect happiness; but they did find that employment, compared with unemployment, was correlated with happiness. Also, they found that TV watching was inversely related to happiness, an effect observed in Tkach and Lyubomirsky's (2006) study mentioned above. Robinson and Martin propose that this effect is observed because passive leisure like TV watching reduces the amount of time spent on activities that may be more beneficial in the long term like exercising, or keeping in contact with friends. Robinson and Martin observed that unhappy people were more likely to feel they either had too much time on their hands or were rushed for time, whereas happy people generally felt neither.
{{Hide in print|
===Increasing Happiness: Quiz Question===
&lt;quiz display=simple&gt;
{Which of the following have '''NOT''' been shown to increase happiness? (Hint: Choose more than one)
|type=&quot;[]&quot;}
+ Watching TV
- Spending time with relatives
+ Subdued expression of emotions
- Exercise
&lt;/quiz&gt;
}}
==Conclusion==
It is important to note that much of the happiness research has been purely correlational in nature, and as such causal connections can not be made. It is possible that activities like socialising with friends increases happiness, but it is also possible that happier people are more likely to engage in social activities; and it could also be that the causal factor is a third variable. When determining which methods are best for increasing or sustaining happiness, research that is experimental in nature will elucidate the most accurate strategies. Evidence from the experimental designs in the cognitive models above suggest that the following will help to increase happiness. Firstly, it is important to make decisions that will be beneficial in the long-run as opposed to succumbing to instant gratification (Lyubomirsky &amp; Tucker). In self-reflection it is beneficial to have a genuine and sincere curiosity about ones-self, rather than to self-ruminate in an anxious and self-criticising way (Trapnell &amp; Campbell, 1999). It is important to have fair and honest view of ones self, ones abilities and ones situation as to avoid discrepancies in actual-ideal and actual-ought selves. And finally, it is beneficial to take some ''constructive'' criticism from an adverse event, and to have a laugh at yourself every now and again.

==References==
{{Hanging indent|1=
Davern, M., Cummins, R., &amp; Stokes, M. (2007). Subjective Wellbeing as an affective-cognitive construct. ''Journal of Happiness Studies, 8'', 429-449. doi:10.1007/s10902-007-9066-1

Diener, E., &amp; Ryan, K. (2009). Subjective well-being: a general overview. ''South African Journal of Psychology, 39'', 391-406. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=45828528&amp;site=ehost-live

Elliott, I. &amp; Coker, S. (2008). Independent self-construal, self-reflectoin, and self-rumination: A path model for predicting happiness. ''Australian Journal of Psychology, 60'', 3, 127-134. doi: 10.1080/00049530701447368

Ersner-Hershfield, H., Mikels, J., Sullivan, S., &amp; Carstensen, L. (2008). Poignancy: Mixed emotional experience in the face of meaningful endings. ''Journal of Personality &amp; Social Psychology, 94'', 158-167. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=28025541&amp;site=ehost-live

Garđarsdóttir, R., Dittmar, H., &amp; Aspinall, C. (2009). It's not the money, it's the quest for a happier self: The role of happiness and success motives in the link between financial goals and subjective well-being. ''Journal of Social and Clinical Psychology, 28'', 1100-1127. doi:10.1521/jscp.2009.28.9.1100

Griffin, J. (2007). What do happiness studies study? ''Journal of Happiness Studies, 8'', 139-148. doi:10.1007/s10902-006-9007-4

Hotard, S., McFatter, R., McWhirter, R., &amp; Stegall, M. (1989). Interactive effects of extraversion, neuroticism, and social relationships on subjective well-being. ''Journal of Personality and Social Psychology, 57'', 321-331. doi:10.1037/0022-3514.57.2.321

Jacobsen, B. (2007). What is happiness? Existential Analysis. ''Journal of the Society for Existential Analysis, 18'', 39-50. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=24478440&amp;site=ehost-live

Lacey, H., Smith, D., &amp; Ubel, P. (2006). Hope I die before I get old: Mispredicting happiness across the adult lifespan. ''Journal of Happiness Studies, 7'', 167-182. doi:10.1007/s10902-005-2748-7

Lyubomirsky, S. (2001). Why are some people happier than others? ''American Psychologist, 56'', 239. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=4298312&amp;site=ehost-live

Lyubomirsky, S., King, L., &amp; Diener, E. (2005). The benefits of frequent positive affect: Does happiness lead to success?. ''Psychological Bulletin, 131'', 803-855. doi:10.1037/0033-2909.131.6.803.

Lyubomirsky, S., &amp; Ross, L. (1997). Hedonic consequences of social comparison: A contrast of happy and unhappy people. ''Journal of Personality &amp; Social Psychology, 73'', 1141-1157. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=76579&amp;site=ehost-live

Lyubomirsky, S., &amp; Tucker, K. (1998). Implications of  individual differences in subjective happiness for perceiving, interpreting, and thinking about life events. ''Motivation &amp; Emotion, 22'', 155-186. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=11307474&amp;site=ehost-live

McDaniel, B., &amp; Grice, J. (2008). Predicting psychological well-being from self-discrepancies: A comparison of idiographic and nomothetic measures. ''Self &amp; Identity, 7'', 243-261. doi:10.1080/15298860701438364

McMahon, D. (2010). What does the ideal of happiness mean?. ''Social Research, 77'', 469-490. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=52721724&amp;site=ehost-live

Morin, A. (2002). Do you “self-reflect” or “self-ruminate”? ''Science and Consciousness Review, 1'', 1-5. Retrieved from http://cogprints.org/3788/

Nesse, R. M. (2005). Natural selection and the elusiveness of happiness. In F. A. Huppert, N. Baylis, &amp; Keverne, B. (Eds.), ''The science of well-being'' (pp. 3-34). Oxford: Oxford University Press.

North, R., Holahan, C., Moos, R., &amp; Cronkite, R. (2008). Family support, family income, and happiness: A 10-year perspective. ''Journal of Family Psychology, 22'', 475-483. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=32602932&amp;site=ehost-live

Robinson, J., &amp; Martin, S. (2008). What do happy people do? ''Social Indicators Research, 89'', 565-571. doi:10.1007/s11205-008-9296-6.

Tkach, C., &amp; Lyubomirsky, S. (2006). How do people pursue happiness?: Relating personality, happiness-increasing strategies, and well-being. ''Journal of Happiness Studies, 7'', 183-225. doi:10.1007/s10902-005-4754-1

Trapnell, P., &amp; Campbell, J. (1999). Private Self-Consciousness and the Five-Factor Model of Personality: Distinguishing Rumination From Reflection. ''Journal of Personality &amp; Social Psychology, 76'', 284-304. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&amp;db=aph&amp;AN=1636560&amp;site=ehost-live

Veenhoven, R. (2008). Healthy happiness: effects of happiness on physical health and the consequences for preventive health care. ''Journal of Happiness Studies, 9'', 449-469. doi:10.1007/s10902-006-9042-1}}

[[Category:Motivation and emotion/Book/2010]]
[[Category:Motivation and emotion/Book/Happiness]]</text>
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  <page>
    <title>Talk:Bible/King James/Documentary Hypothesis/Genesis</title>
    <ns>1</ns>
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      <parentid>2623430</parentid>
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      <contributor>
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        <id>2928717</id>
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      <comment>/* Jahwist/Elohist */ Reply</comment>
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      <text bytes="3370" xml:space="preserve">I was wondering what source was used to compile this, because I have found that some of the isolated sources don't reflect this article, and found contradictory information elsewhere on the web.--[[Special:Contributions/86.165.76.14|86.165.76.14]] 08:39, 4 January 2011 (UTC)
: I endorse this question completely; the only complete assignments I know of are Friedman, for which this would be a copyright violation; Reuss for which it wouldn't but you would have to read French to figure him out; and Astruc who only assigned Genesis.  So which is it? [[Special:Contributions/71.163.117.143|71.163.117.143]] ([[User talk:71.163.117.143|discuss]]) 03:15, 1 February 2015 (UTC)
::It seems like people are editing this based on their personal preference. There are no citations. Something needs to be done to maintain consistency.--[[User:Jcvamp|Jcvamp]] ([[User talk:Jcvamp|discuss]] • [[Special:Contributions/Jcvamp|contribs]]) 18:42, 15 December 2016 (UTC)
:::This is fairly old, but Joel Baden (a respected source critical bible scholar) is currently giving a verse by verse analysis of the Pentateuch according to the contemporary neo-DH. I'm planning on using this as a guide to update this article, but since discussion exists, thought I should bring up the idea here first. [[User:Huz and Buz|Huz and Buz]] ([[User talk:Huz and Buz|discuss]] • [[Special:Contributions/Huz and Buz|contribs]]) 20:09, 5 October 2021 (UTC)

The choice of colors for the different sources is AWFUL: I could barely tell the difference at times.--[[User:Mwidunn|Mwidunn]] ([[User talk:Mwidunn|discuss]] • [[Special:Contributions/Mwidunn|contribs]]) 03:59, 13 July 2014 (UTC)mwidunn

== Jahwist/Elohist ==

The Jahwist and Elohist sources are no longer treated as substantial by scholars and are today treated as a whole source, so I think Wikimedia should do the same. [[User:GOLDIEM J|GOLDIEM J]] ([[User talk:GOLDIEM J|discuss]] • [[Special:Contributions/GOLDIEM J|contribs]]) 10:59, 16 March 2024 (UTC)

:I used the source divisions provided by Joel Baden because they are detailed and contemporary. He explains why he believes in a separate E source in his book &quot;The Composition of the Pentateuch: Renewing the Documentary Hypothesis.&quot; There's a lot of disagreement on the sources obviously, so maybe in consideration of this there should be a way to see the JE source as one document, like you currently can with J, E, P, and D. [[User:Huz and Buz|Huz and Buz]] ([[User talk:Huz and Buz|discuss]] • [[Special:Contributions/Huz and Buz|contribs]]) 18:22, 1 April 2024 (UTC)
::Hello! I've read Who Wrote The Bible by Friedman and The Composition Of The Pentateuch by Baden. I hoped Composition would have source identification by verse, but it doesn't. Where can I find Baden's work? I have seen the info here, but I'd like to know the source. I checked some of the Twitter references, and didn't see what I was looking for.     Thanks, Scott [[User:LoreOfYore|LoreOfYore]] ([[User talk:LoreOfYore|discuss]] • [[Special:Contributions/LoreOfYore|contribs]]) 17:27, 27 April 2024 (UTC)
:::It's a little hard to navigate, but the pinned tweet thread on Joel Baden's account contains all of his source identifications, verse by verse. [[User:Huz and Buz|Huz and Buz]] ([[User talk:Huz and Buz|discuss]] • [[Special:Contributions/Huz and Buz|contribs]]) 21:50, 2 May 2024 (UTC)</text>
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    <title>File:Paragogy-final.pdf</title>
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      <text bytes="886" xml:space="preserve">== Summary ==
{{Information
|Description=
This paper describes a new theory of peer-to-peer learning
and teaching that we call &quot;paragogy&quot;. Paragogy's principles were developed
by adapting the Knowles's principles of andragogy to peer-based
learning contexts. Paragogy addresses the challenge of peer-producing a
useful and supportive context for self-directed learning.
The concept of paragogy can inform the design and application of learning
analytics to enhance both individual and organization learning. In
particular, we consider the role of learner profiles for goal-setting and
self-monitoring, and the further role of analytics in designing enhanced
peer tutoring systems.
|Source={{own}}
|Date=2011
|Author=Joe Cornelli and Charles Jeffrey Danoff ([[User:Charles Jeffrey Danoff]])
|Permission=
}}

== Licensing ==
{{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
[[Category:WikiJournal]]</text>
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    <title>EuroLex/F/Toilet</title>
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      <comment>added [[Category:Toilets]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="4220" xml:space="preserve">*'''Original language''': French
*'''Original form and meaning''': ''toilette''  'lavatory, bathroom, to dress up'


(Note: If the status is not specifically indicated then the word is stylistically neutral and generally used; if earlier meaning and status equals current use the former may be expressed by writing &quot;dito&quot;. Cf. also the project [[ELiX Wiki:Projects/EuroLex|guidelines]].)


&lt;table border=1&gt;
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&lt;td width=10%&gt;'''Language'''
&lt;td width=15%&gt;'''Form'''
&lt;td width=10%&gt;'''Date of Borrowing (and Obsolescence) '''
&lt;td width=25%&gt;'''Current Meaning and Status'''
&lt;td width=25%&gt;'''Earlier Meanings and Statusses'''
&lt;td width=15%&gt;'''Source'''
&lt;tr&gt;
	&lt;td&gt;Catalan
	&lt;td&gt;''...''
	&lt;td&gt;...
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	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Croatian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Czech
	&lt;td&gt;''toaleta; toaletní''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Danish
	&lt;td&gt;''toilette; toilet''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Dutch
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;English
	&lt;td&gt;''toilet''
	&lt;td&gt;16th c.
	&lt;td&gt;1. 'a cloth cover for a dressing table (now usually calles a 'toilet-cover'), 2. 'the articles required or used in dressing, the furniture of the toilet-table' (collective), 3. 'the table on which these articles are placed', 4. 'the action or process of dressing, or, more recently, of washing and grooming', 5. 'manner or style of dressing, dress, cosume,'get-up', gown', 6. 'a dressing-room, in U.S. esp. a dressing-room furnished with bathing facilities, hence a bath-room, a lavatory, (contextually), a lavatory bowl or pedestral, a room or cubicle containing a lavatory'
	&lt;td&gt;'cover or bag for clothes'
	&lt;td&gt;OED, http://www.etymonline.com
&lt;tr&gt;
	&lt;td&gt;Estonian
	&lt;td&gt;''tualett(ruum)''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Finnish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;French
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Frisian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;German
	&lt;td&gt;''Toilette''
	&lt;td&gt;18th c.
	&lt;td&gt;1. 'fine women's clothing' (dial.), 2. 'lavatory' (dial.), 3. 'to dress oneself' (dial.), 4. 'clothes' (dial.), 5.'water closet' (dial.)
	&lt;td&gt;'...'
	&lt;td&gt;Birken-Silvermann 2003: 139
&lt;tr&gt;
	&lt;td&gt;Hungarian
	&lt;td&gt;''toalett''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Irish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Italian
	&lt;td&gt;''toilette, toaletta''
	&lt;td&gt;1805, 1825
	&lt;td&gt;1. 'dressing table, to dress oneself' (dial.), 2. 'new clothes, lavatory' (dial.), 3. 'mirror' (dial.)
	&lt;td&gt;'...'
	&lt;td&gt;Birken-Silvermann 2003: 139
&lt;tr&gt;
	&lt;td&gt;Latvian
	&lt;td&gt;''tualete''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Lithuanian
	&lt;td&gt;''tualetas''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Maltese
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Norwegian
	&lt;td&gt;''toalett''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Polish
	&lt;td&gt;''toaleta''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Portuguese
	&lt;td&gt;''Brazil: toalete''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Rumantsch
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Slovak
	&lt;td&gt;''toaleta; toaletný''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Slovenian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Spanish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Swedish
	&lt;td&gt;''toalett''
	&lt;td&gt;...
	&lt;td&gt;'lavatory'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;/table&gt;

=== Annotations ===
Etymology: original English meaning: &quot;cover or bag for clothes&quot;, from MF ''toilette'' &quot;a cloth, bag for clothes&quot; diminutive of ''toile'' &quot;cloth, net&quot;. Sense evolution is to &quot;act or process of dressing&quot; (1681); then &quot;a dressing room&quot; (1819), especially one with a lavatory attached; then &quot;lavatory or porcelain plumbing fixture&quot; (1895), an AmE euphemistic use. ''Toilet paper'' is attested from 1884. ''Toilet training'' is recorded from 1940. 

Source: http://www.etymonline.com

=== Information on Other Languages ===
Estonian: WC

Finnish: WC

Portuguese: privada, banheiro

Slovenian: stranišče

Spanish: lavabo

[[Category:EuroLex]]
[[Category:Gallicism]]
[[Category:Toilets]]</text>
      <sha1>lfn402r4iwj6rwoy82mfm001j09vzdk</sha1>
    </revision>
  </page>
  <page>
    <title>EuroLex/F/Toilet water</title>
    <ns>0</ns>
    <id>111666</id>
    <revision>
      <id>2624955</id>
      <parentid>729384</parentid>
      <timestamp>2024-05-03T08:07:21Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>added [[Category:Toilets]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2975" xml:space="preserve">*'''Original language''': French
*'''Original form and meaning''': ''eau de toilette'' - 1. toilet water (a scented liquid with a high alcohol content used in bathing or applied as a skin freshener, similar to cologne)


(Note: If the status is not specifically indicated then the word is stylistically neutral and generally used; if earlier meaning and status equals current use the former may be expressed by writing &quot;dito&quot;. Cf. also the project [[ELiX Wiki:Projects/EuroLex|guidelines]].)


&lt;table border=1&gt;
&lt;tr&gt;
&lt;td width=10%&gt;'''Language'''
&lt;td width=15%&gt;'''Form'''
&lt;td width=10%&gt;'''Date of Borrowing (and Obsolescence) '''
&lt;td width=25%&gt;'''Current Meaning and Status'''
&lt;td width=25%&gt;'''Earlier Meanings and Statusses'''
&lt;td width=15%&gt;'''Source'''
&lt;tr&gt;
	&lt;td&gt;Catalan
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Croatian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Czech
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Danish
	&lt;td&gt;''eau de toilette''
	&lt;td&gt;...
	&lt;td&gt;'meaning 1'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Dutch
	&lt;td&gt;''eau de toilet''
	&lt;td&gt;...
	&lt;td&gt;'meaning 1'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;English
	&lt;td&gt;''toilet water (loan-translation)''
	&lt;td&gt;19 c
	&lt;td&gt;'meaning 1'
	&lt;td&gt;'...'
	&lt;td&gt;http://dictionary.reference.com
&lt;tr&gt;
	&lt;td&gt;Estonian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Finnish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;French
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Frisian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;German
	&lt;td&gt;''Eau de Toilette''
	&lt;td&gt;...
	&lt;td&gt;'meaning 1'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Hungarian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Irish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Italian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Latvian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Lithuanian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Maltese
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Norwegian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Polish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Portuguese
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Rumantsch
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Slovak
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Slovenian
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Spanish
	&lt;td&gt;''...''
	&lt;td&gt;...
	&lt;td&gt;'...'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;tr&gt;
	&lt;td&gt;Swedish
	&lt;td&gt;''eau-de-toilette''
	&lt;td&gt;...
	&lt;td&gt;'meaning 1'
	&lt;td&gt;'...'
	&lt;td&gt;...
&lt;/table&gt;

=== Annotations ===
Etymology of  [[Toilet (EuroLex)]].

=== Information on Other Languages ===
'''loan-translations''':

German: Toilettenwasser

Lithuanian: tualetinis vanduo (?)

Polish: woda toaletowa

Slovenian: toaletna voda


[[Category:EuroLex]]
[[Category:Gallicism]]
[[Category:Toilets]]</text>
      <sha1>kg3hb9molu3irjc2lzqcvdqfbr6xr7b</sha1>
    </revision>
  </page>
  <page>
    <title>Template:Music symbols/doc</title>
    <ns>10</ns>
    <id>114821</id>
    <revision>
      <id>2624580</id>
      <parentid>2582615</parentid>
      <timestamp>2024-05-02T12:08:16Z</timestamp>
      <contributor>
        <ip>110.22.161.29</ip>
      </contributor>
      <comment>/* Notes and rests */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16238" xml:space="preserve">{{documentation subpage}}
{{See also|Template:Music topics}}
&lt;!--Hopefully the inline style attribute will be replace by a class reference to &quot;accidental&quot; in [[MediaWiki:Common.css]]--&gt;

{{tl|Music symbols}} renders Western music notation of various types into Wikipedia and improves cross-browser support for music symbols.

 1     2     3     4     5     6     7     8
| 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 |

==Accidentals==
The template correctly renders Unicode sharps ({{music symbols|sharp}}), flats ({{music symbols|flat}}), and natural signs ({{music symbols|natural}}) in [[w:Internet Explorer]] which would otherwise display [[w:Mojibake|empty squares]] unless a full Unicode font is chosen in its Preferences. The choice of fonts also improves the rendering in other browsers on [[w:Microsoft Windows]] such as [[w:Mozilla Firefox]]. See the tables below to compare the results in your current browser. The template makes use of [[w:Scalable Vector Graphics|SVG]] to display double flat ({{music symbols|doubleflat}}), double sharp ({{music symbols|doublesharp}}), and microtonal signs since the corresponding Unicode characters are not widely supported.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;entity 
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;result 
! style=&quot;width:18em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Flat
| &lt;code&gt;&amp;amp;#x266d;&lt;/code&gt; || &amp;#x266d;
| {{tlc|music symbols|flat}}&lt;br/&gt;''or'' {{tlc|music symbols|b}}&lt;br/&gt;''or'' {{tlc|music symbols|♭}} ||  {{music symbols|flat}}
|-
! Natural
| &lt;code&gt;&amp;amp;#x266e;&lt;/code&gt;  || &amp;#x266e; 
| {{tlc|music symbols|natural}}&lt;br/&gt;''or'' {{tlc|music symbols|♮}} || {{music symbols|natural}}
|-
! Sharp
| &lt;code&gt;&amp;amp;#x266f;&lt;/code&gt;  || &amp;#x266f;
| {{tlc|musi symbolsc|sharp}}&lt;br/&gt;''or'' {{tlc|music symbols|#}}&lt;br/&gt;''or'' {{tlc|music symbols|♯}} || {{music symbols|sharp}}
|-
! Double&lt;br/&gt;flat
| &lt;code&gt;&amp;amp;#x1D12B;&lt;/code&gt;  || &amp;#x1D12B;
| {{tlc|music symbols|doubleflat}}&lt;br/&gt;''or'' {{tlc|music symbols|bb}}&lt;br/&gt;''or'' {{tlc|music symbols|𝄫}} || {{music symbols|doubleflat}}
|-
! Double sharp
| &lt;code&gt;&amp;amp;#x1D12A;&lt;/code&gt;  || &amp;#x1D12A;
| {{tlc|music symbols|doublesharp}}&lt;br/&gt;''or'' {{tlc|music symbols|##}}&lt;br/&gt;''or'' {{tlc|music symbols|x}}&lt;br/&gt;''or'' {{tlc|music symbols|𝄪}} || {{music symbols|doublesharp}}
|-
! Half&lt;br/&gt;flat
| &lt;code&gt;&amp;amp;#x1D133;&lt;/code&gt;  || &amp;#x1D133;
| {{tlc|music symbols|halfflat}}&lt;br/&gt;''or'' {{tlc|music symbols|d}} || {{music symbols|halfflat}}
|-
! Half&lt;br/&gt;sharp
| &lt;code&gt;&amp;amp;#x1D132;&lt;/code&gt;  || &amp;#x1D132;
| {{tlc|music symbols|halfsharp}}&lt;br/&gt;''or'' {{tlc|music symbols|t}} || {{music symbols|halfsharp}}
|-
! Flat&lt;br/&gt;stroke
| &lt;code&gt;&amp;amp;#x1D133;&lt;/code&gt;  || &amp;#x1D133;
| {{tlc|music symbols|flatstroke}} || {{music symbols|flatstroke}}
|-
! Three quarter flat
| &lt;code&gt; &lt;/code&gt;  || 
| {{tlc|music symbols|threequarterflat}}&lt;br/&gt;''or'' {{tlc|music symbols|db}} || {{music symbols|threequarterflat}}
|-
! Three quarter sharp
| &lt;code&gt; &lt;/code&gt;  || 
| {{tlc|music symbols|threequartersharp}}&lt;br/&gt;''or'' {{tlc|music symbols|#t}} || {{music symbols|threequartersharp}}
|-
! Double&lt;br/&gt;flat&lt;br/&gt;stroke
| &lt;code&gt; &lt;/code&gt; ||
| {{tlc|music symbols|doubleflatstroke}} || {{music symbols|doubleflatstroke}}
|}

;Sample text
The C{{music symbols|#}} crops up very early in Beethoven's Symphony No. 3 in E{{music symbols|b}}.

==Notes and rests==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;entity 
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;result 
! style=&quot;width:18em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Whole&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15d;&lt;/code&gt; || &amp;#x1d15d;
| {{tlc|music symbols|wholenote}}&lt;br/&gt;''or'' {{tlc|music symbols|whole}}&lt;br/&gt;''or'' {{tlc|music symbols|semibreve}} || {{music symbols|whole}}
|-
! Half&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15e;&lt;/code&gt; || &amp;#x1d15e;
| {{tlc|music symbols|halfnote}}&lt;br/&gt;''or'' {{tlc|music symbols|half}}&lt;br/&gt;''or'' {{tlc|music symbols|minim}} || {{music symbols|half}}
|-
! Quarter&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15f;&lt;/code&gt; || &amp;#x1d15f;
| {{tlc|music symbols|quarternote}}&lt;br/&gt;''or'' {{tlc|music symbols|quarter}}&lt;br/&gt;''or'' {{tlc|music symbols|crotchet}} || {{music symbols|quarter}}
|-
! Eighth&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d160;&lt;/code&gt; || &amp;#x1d160;
| {{tlc|music symbols|eighthnote}}&lt;br/&gt;''or'' {{tlc|music symbols|eighth}}&lt;br/&gt;''or'' {{tlc|music symbols|quaver}} || {{music symbols|eighth}}
|-
! Beamed&lt;br/&gt;eighth&lt;br/&gt;notes
| ||
| {{tlc|music symbols|eighthnotebeam}}&lt;br/&gt;''or'' {{tlc|music symbols|eighthbeam}}&lt;br/&gt;''or'' {{tlc|music symbols|quaverbeam}} || {{music symbols|eighthbeam}}
|-
! Sixteenth&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d161;&lt;/code&gt; || &amp;#x1d161;
| {{tlc|music symbols|sixteenthnote}}&lt;br/&gt;''or'' {{tlc|music symbols|sixteenth}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaver}} || {{music symbols|sixteenth}}
|-
! Beamed&lt;br/&gt;sixteenth&lt;br/&gt;notes
| ||
| {{tlc|music symbols|sixteenthnotebeam}}&lt;br/&gt;''or'' {{tlc|music symbols|sixteenthbeam}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaverbeam}} || {{music symbols|sixteenthbeam}}
|-
! Thirty-&lt;br/&gt;second&lt;br/&gt;note
| ||
| {{tlc|music symbols|thirty-secondnote}}&lt;br/&gt;''or'' {{tlc|music symbols|thirty-second}}&lt;br/&gt;''or'' {{tlc|music symbols|demisemiquaver}} || {{music symbols|thirty-second}}
|-
! Dot
| . || .
| {{tlc|music symbols|dot}} || {{music symbols|dot}}
|-
! Dotted quarter&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d161;&lt;/code&gt;. || &amp;#x1d161;.
| {{tlc|music symbols|dottedquarter}}&lt;br/&gt;''or'' {{tlc|music symbols|dottedcrotchet}} || {{music symbols|dottedquarter}}
|-
! Dotted half&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15e;&lt;/code&gt;. || &amp;#x1d15e;.
| {{tlc|music symbols|dottedhalf}}&lt;br/&gt;''or'' {{tlc|music symbols|dottedminim}} || {{music symbols|dottedhalf}}
|-
! Whole&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13b;&lt;/code&gt; || &amp;#x1d13b;
| {{tlc|music symbols|wholerest}}&lt;br/&gt;''or'' {{tlc|music symbols|semibreverest}} || {{music symbols|wholerest}}
|-
! Half&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13c;&lt;/code&gt; || &amp;#x1d13c;
| {{tlc|music symbols|halfrest}}&lt;br/&gt;''or'' {{tlc|music symbols|minimrest}} || {{music symbols|halfrest}}
|-
! Quarter&lt;br/&gt;rest bombaclat

| &lt;code&gt;&amp;amp;#x1d13d;&lt;/code&gt; || &amp;#x1d13d;
| {{tlc|music symbols|quarterrest}}&lt;br/&gt;''or'' {{tlc|music symbols|crotchetrest}} || {{music symbols|quarterrest}}
|-
! Eighth&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13e;&lt;/code&gt; || &amp;#x1d13e;
| {{tlc|music symbols|eighthrest}}&lt;br/&gt;''or'' {{tlc|music symbols|quaverrest}} || {{music symbols|eighthrest}}
|-
! Sixteenth&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13f;&lt;/code&gt; || &amp;#x1d13f;
| {{tlc|music symbols|sixteenthrest}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaverrest}} || {{music symbols|sixteenthrest}}
|-
! Thirty-&lt;br/&gt;second&lt;br/&gt;rest
| ||
| {{tlc|music symbols|thirtysecondrest}}&lt;br/&gt;''or'' {{tlc|music symbols|demisemiquaverrest}} || {{music symbols|thirtysecondrest}}
|}

Some browsers and typefaces support &lt;code&gt;&amp;amp;#x2669;&lt;/code&gt; (&amp;#x2669;) and &lt;code&gt;&amp;amp;#x266a;&lt;/code&gt; (&amp;#x266a;) for quarter and eighth notes, as well as &lt;code&gt;&amp;amp;#x266b;&lt;/code&gt; (&amp;#x266b;) and &lt;code&gt;&amp;amp;#x266c;&lt;/code&gt; (&amp;#x266c;) for beamed eighth-note and sixteenth-note pairs respectively, but since the display of these characters does not match any of the other (non-supported) notes and rests, this template does not use these characters.

;Sample text
In place of the single whole note ({{music symbols|semibreve}}), Chopin writes {{music symbols|eighthrest}} {{music symbols|halfnote}} {{music symbols|quarter}} {{music symbols|eighth}}, completely changing the profile of the music.

==Clefs==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Treble clef&lt;br /&gt;''or'' G-clef
| {{tlc|music symbols|treble}}&lt;br/&gt;''or'' {{tlc|music symbols|trebleclef}}&lt;br/&gt;''or'' {{tlc|music symbols|gclef}} || {{music symbols|trebleclef}}
|-
! Alto&lt;br/&gt;clef
| {{tlc|music symbols|alto}}&lt;br/&gt;''or'' {{tlc|music symbols|altoclef}} || {{music symbols|altoclef}}
|-
! Tenor&lt;br/&gt;clef
| {{tlc|music symbols|tenor}}&lt;br/&gt;''or'' {{tlc|music symbols|tenorclef}} || {{music symbols|tenorclef}}
|-
! C-clef
| {{tlc|music symbols|cclef}}  || {{music symbols|cclef}}
|-
! Bass clef&lt;br /&gt;''or'' F-clef
| {{tlc|music symbols|bass}}&lt;br/&gt;''or'' {{tlc|music symbols|bassclef}}&lt;br /&gt;''or'' {{tlc|music symbols|fclef}}  || {{music symbols|bassclef}}
|-
! Neutral clef
| {{tlc|music symbols|neutral}}&lt;br/&gt;''or'' {{tlc|music symbols|neutralclef}}&lt;br /&gt; || {{music symbols|neutralclef}}
|}

Note that there is no graphical distinction between treble clef and G-clef; alto clef, tenor clef and C-clef; bass clef and F-clef. The names preserve a difference in meaning and make the caption text (for screen readers) different.

== Time signatures ==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! common-time
| {{tlc|music symbols|common-time}} || {{music symbols|common-time}}
|-
! cut-time
| {{tlc|music symbols|cut-time}}&lt;br/&gt;''or'' {{tlc|music symbols|alla-breve}} || {{music symbols|cut-time}}
|-
! 2/4
| {{tlc|music symbols|time|2|4}} || {{music symbols|time|2|4}}
|-
! 3/4
| {{tlc|music symbols|time|3|4}} || {{music symbols|time|3|4}}
|-
! 6/8
| {{tlc|music symbols|time|6|8}} || {{music symbols|time|6|8}}
|-
! 9/8
| {{tlc|music symbols|time|9|8}} || {{music symbols|time|9|8}}
|-
! 12/8
| {{tlc|music symbols|time|12|8}} || {{music symbols|time|12|8}}
|}
For a ''general'' time signature, use &lt;nowiki&gt;{{music symbols|time|&lt;top number&gt;|&lt;bottom number&gt;}}&lt;/nowiki&gt;. This makes use of [[Template:Time signature]], which should not be used on its own.

==Scale degrees==
Scale degrees are often represented as Arabic numerals with a hat on them and thus the root of a scale is &lt;sub&gt;{{music symbols|scale|1}}&lt;/sub&gt;.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Scale degree
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! 1
| {{tlc|music symbols|scale|1}} || {{music symbols|scale|1}}
|-
! 2
| {{tlc|music symbols|scale|2}} || {{music symbols|scale|2}}
|-
! 3
| {{tlc|music symbols|scale|3}} || {{music symbols|scale|3}}
|-
! 4
| {{tlc|music symbols|scale|4}} || {{music symbols|scale|4}}
|-
! 5
| {{tlc|music symbols|scale|5}} || {{music symbols|scale|5}}
|-
! 6
| {{tlc|music symbols|scale|6}} || {{music symbols|scale|6}}
|-
! 7
| {{tlc|music symbols|scale|7}} || {{music symbols|scale|7}}
|-
! 8
| {{tlc|music symbols|scale|8}} || {{music symbols|scale|8}}
|-
! 9
| {{tlc|music symbols|scale|9}} || {{music symbols|scale|9}}
|}

;Sample text
A descending tetrachord could be written as &lt;sub&gt;{{music symbols|scale|4}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|3}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|2}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|1}}&lt;/sub&gt;.

==Chord symbols==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:9em;&quot; | Chord
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Diminished
| {{tlc|music symbols|dim}}&lt;br/&gt;''or'' {{tlc|music symbols|dimdeg}} || {{music symbols|dim}}&lt;br/&gt;''or'' {{music symbols|dimdeg}}
|-
! Half-diminished
| {{tlc|music symbols|halfdim}}&lt;br/&gt;''or'' {{tlc|music symbols|dimslash}} || {{music symbols|halfdim}}
|-
! Augmented
| {{tlc|music symbols|+}}&lt;br/&gt;''or'' {{tlc|music symbols|aug}} || {{music symbols|aug}}
|-
! Major
| {{tlc|music symbols|delta}}&lt;br/&gt;''or'' {{tlc|music symbols|major}} || {{music symbols|major}}
|}

;Sample text
vii{{music symbols|dim}} becomes vii{{music symbols|dimslash}} in B{{music symbols|flat}} minor by raising the G{{music symbols|b}} to G{{music symbols|natural}}.

III becomes III{{music symbols|aug}} in G{{music symbols|#}} minor by raising the F{{music symbols|#}} to F{{music symbols|x}}.

The F{{music symbols|#}}{{music symbols|major}}&lt;sup&gt;7&lt;/sup&gt; is used to great effect in the last measure of the piece.

==Key signatures==
Key signatures must be typed in using the names of the articles themselves on keys. Note that there is no visual difference between the major and minor key signatures, but the alt text is different.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! Sharps/Flats
! style=&quot;width:9em;&quot; | Key signature (major)
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
! style=&quot;width:9em;&quot; | Key signature (minor)
! style=&quot;width:16em; | Template&lt;br/&gt;text
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! {{font|color=blue|7{{music symbols|#}}}}
! [[C-sharp major|C{{music symbols|#}} major]]
| {{tlc|music symbols|c-sharp major}} || {{music symbols|c-sharp major}}
! [[A-sharp minor|A{{music symbols|#}} minor]]
| {{tlc|music symbols|a-sharp minor}} || {{music symbols|a-sharp minor}}
|-
! {{font|color=blue|6{{music symbols|#}}}}
! [[F-sharp major|F{{music symbols|#}} major]]
| {{tlc|music symbols|f-sharp major}} || {{music symbols|f-sharp major}}
! [[D-sharp minor|D{{music symbols|#}} minor]]
| {{tlc|music symbols|d-sharp minor}} || {{music symbols|d-sharp minor}}
|-
! {{font|color=blue|5{{music symbols|#}}}}
! [[B major]]
| {{tlc|music symbols|b major}} || {{music symbols|b major}}
! [[G-sharp minor|G{{music symbols|#}} minor]]
| {{tlc|music symbols|g-sharp minor}} || {{music symbols|g-sharp minor}}
|-
! {{font|color=blue|4{{music symbols|#}}}}
! [[E major]]
| {{tlc|music symbols|e major}} || {{music symbols|e major}}
! [[C-sharp minor|C{{music symbols|#}} minor]]
| {{tlc|music symbols|c-sharp minor}} || {{music symbols|c-sharp minor}}
|-
! {{font|color=blue|3{{music symbols|#}}}}
! [[A major]]
| {{tlc|music symbols|a major}} || {{music symbols|a major}}
! [[F-sharp minor|F{{music symbols|#}} minor]]
| {{tlc|music symbols|f-sharp minor}} || {{music symbols|f-sharp minor}}
|-
! {{font|color=blue|2{{music symbols|#}}}}
! [[D major]]
| {{tlc|music symbols|d major}} || {{music symbols|d major}}
! [[B minor]]
| {{tlc|music symbols|b minor}} || {{music symbols|b minor}}
|-
! {{font|color=blue|1{{music symbols|#}}}}
! [[G major]]
| {{tlc|music symbols|g major}} || {{music symbols|g major}}
! [[E minor]]
| {{tlc|music symbols|e minor}} || {{music symbols|e minor}}
|-
! {{font|color=green|0}}
! [[C major]]
| {{tlc|music symbols|c major}} || {{music symbols|c major}}
! [[A minor]]
| {{tlc|music symbols|a minor}} || {{music symbols|a minor}}
|-
! {{font|color=red|1{{music symbols|b}}}}
! [[F major]]
| {{tlc|music symbols|f major}} || {{music symbols|f major}}
! [[D minor]]
| {{tlc|music symbols|d minor}} || {{music symbols|d minor}}
|-
! {{font|color=red|2{{music symbols|b}}}}
! [[B-flat major|B{{music symbols|b}} major]]
| {{tlc|music symbols|b-flat major}} || {{music symbols|b-flat major}}
! [[G minor]]
| {{tlc|music symbols|g minor}} || {{music symbols|g minor}}
|-
! {{font|color=red|3{{music symbols|b}}}}
! [[E-flat major|E{{music symbols|b}} major]]
| {{tlc|music symbols|e-flat major}} || {{music symbols|e-flat major}}
! [[C minor]]
| {{tlc|music symbols|c minor}} || {{music symbols|c minor}}
|-
! {{font|color=red|4{{music symbols|b}}}}
! [[A-flat major|A{{music symbols|b}} major]]
| {{tlc|music symbols|a-flat major}} || {{music symbols|a-flat major}}
! [[F minor]]
| {{tlc|music symbols|f minor}} || {{music symbols|f minor}}
|-
! {{font|color=red|5{{music symbols|b}}}}
! [[D-flat major|D{{music symbols|b}} major]]
| {{tlc|music symbols|d-flat major}} || {{music symbols|d-flat major}}
! [[B-flat minor|B{{music symbols|b}} minor]]
| {{tlc|music symbols|b-flat minor}} || {{music symbols|b-flat minor}}
|-
! {{font|color=red|6{{music symbols|b}}}}
! [[G-flat major|G{{music symbols|b}} major]]
| {{tlc|music symbols|g-flat major}} || {{music symbols|g-flat major}}
! [[E-flat minor|E{{music symbols|b}} minor]]
| {{tlc|music symbols|e-flat minor}} || {{music symbols|e-flat minor}}
|-
! {{font|color=red|7{{music symbols|b}}}}
! [[C-flat major|C{{music symbols|b}} major]]
| {{tlc|music symbols|c-flat major}} || {{music symbols|c-flat major}}
! [[A-flat minor|A{{music symbols|b}} minor]]
| {{tlc|music symbols|a-flat minor}} || {{music symbols|a-flat minor}}
|}

&lt;includeonly&gt;
[[Category:Music templates|{{BASEPAGENAME}}]]

&lt;/includeonly&gt;</text>
      <sha1>0kcs2fznzba029f81zo3f4xfuj1lias</sha1>
    </revision>
    <revision>
      <id>2624583</id>
      <parentid>2624580</parentid>
      <timestamp>2024-05-02T12:11:46Z</timestamp>
      <contributor>
        <ip>110.22.161.29</ip>
      </contributor>
      <comment>/* Accidentals */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16270" xml:space="preserve">{{documentation subpage}}
{{See also|Template:Music topics}}
&lt;!--Hopefully the inline style attribute will be replace by a class reference to &quot;accidental&quot; in [[MediaWiki:Common.css]]--&gt;

{{tl|Music symbols}} renders Western music notation of various types into Wikipedia and improves cross-browser support for music symbols.

 1     2     3     4     5     6     7     8
| 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 | 𝅝   𝅝   𝅝 |

==Accidentals==
The template correctly renders Unicode sharps ({{music symbols|sharp}}), flats ({{music symbols|flat}}), and natural signs ({{music symbols|natural}}) in [[w:Internet Explorer]] which would otherwise display [[w:Mojibake|empty squares]] unless a full Unicode font is chosen in its Preferences. The choice of fonts also improves the rendering in other browsers on [[w:Microsoft Windows]] such as [[w:Mozilla Firefox]]. See the tables below to compare the results in your current browser. The template makes use of [[w:Scalable Vector Graphics|SVG]] to display double flat ({{music symbols|doubleflat}}), double sharp ({{music symbols|doublesharp}}), and microtonal signs since the corresponding Unicode characters are not widely supported.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;entity 
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;result 
! style=&quot;width:18em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Flat
| &lt;code&gt;&amp;amp;#x266d;&lt;/code&gt; || &amp;#x266d;
| {{tlc|music symbols|flat}}&lt;br/&gt;''or'' {{tlc|music symbols|b}}&lt;br/&gt;''or'' {{tlc|music symbols|♭}} ||  {{music symbols|flat}}
|-
! Natural
| &lt;code&gt;&amp;amp;#x266e;&lt;/code&gt;  || &amp;#x266e; 
| {{tlc|music symbols|natural}}&lt;br/&gt;''or'' {{tlc|music symbols|♮}} || {{music symbols|natural}}
|-
! Sharp  js like kings and devs jawline
| &lt;code&gt;&amp;amp;#x266f;&lt;/code&gt;  || &amp;#x266f;
| {{tlc|musi symbolsc|sharp}}&lt;br/&gt;''or'' {{tlc|music symbols|#}}&lt;br/&gt;''or'' {{tlc|music symbols|♯}} || {{music symbols|sharp}}
|-
! Double&lt;br/&gt;flat
| &lt;code&gt;&amp;amp;#x1D12B;&lt;/code&gt;  || &amp;#x1D12B;
| {{tlc|music symbols|doubleflat}}&lt;br/&gt;''or'' {{tlc|music symbols|bb}}&lt;br/&gt;''or'' {{tlc|music symbols|𝄫}} || {{music symbols|doubleflat}}
|-
! Double sharp
| &lt;code&gt;&amp;amp;#x1D12A;&lt;/code&gt;  || &amp;#x1D12A;
| {{tlc|music symbols|doublesharp}}&lt;br/&gt;''or'' {{tlc|music symbols|##}}&lt;br/&gt;''or'' {{tlc|music symbols|x}}&lt;br/&gt;''or'' {{tlc|music symbols|𝄪}} || {{music symbols|doublesharp}}
|-
! Half&lt;br/&gt;flat
| &lt;code&gt;&amp;amp;#x1D133;&lt;/code&gt;  || &amp;#x1D133;
| {{tlc|music symbols|halfflat}}&lt;br/&gt;''or'' {{tlc|music symbols|d}} || {{music symbols|halfflat}}
|-
! Half&lt;br/&gt;sharp
| &lt;code&gt;&amp;amp;#x1D132;&lt;/code&gt;  || &amp;#x1D132;
| {{tlc|music symbols|halfsharp}}&lt;br/&gt;''or'' {{tlc|music symbols|t}} || {{music symbols|halfsharp}}
|-
! Flat&lt;br/&gt;stroke
| &lt;code&gt;&amp;amp;#x1D133;&lt;/code&gt;  || &amp;#x1D133;
| {{tlc|music symbols|flatstroke}} || {{music symbols|flatstroke}}
|-
! Three quarter flat
| &lt;code&gt; &lt;/code&gt;  || 
| {{tlc|music symbols|threequarterflat}}&lt;br/&gt;''or'' {{tlc|music symbols|db}} || {{music symbols|threequarterflat}}
|-
! Three quarter sharp
| &lt;code&gt; &lt;/code&gt;  || 
| {{tlc|music symbols|threequartersharp}}&lt;br/&gt;''or'' {{tlc|music symbols|#t}} || {{music symbols|threequartersharp}}
|-
! Double&lt;br/&gt;flat&lt;br/&gt;stroke
| &lt;code&gt; &lt;/code&gt; ||
| {{tlc|music symbols|doubleflatstroke}} || {{music symbols|doubleflatstroke}}
|}

;Sample text
The C{{music symbols|#}} crops up very early in Beethoven's Symphony No. 3 in E{{music symbols|b}}.

==Notes and rests==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;entity 
! style=&quot;width:6em;&quot; | Unicode&lt;br/&gt;result 
! style=&quot;width:18em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Whole&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15d;&lt;/code&gt; || &amp;#x1d15d;
| {{tlc|music symbols|wholenote}}&lt;br/&gt;''or'' {{tlc|music symbols|whole}}&lt;br/&gt;''or'' {{tlc|music symbols|semibreve}} || {{music symbols|whole}}
|-
! Half&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15e;&lt;/code&gt; || &amp;#x1d15e;
| {{tlc|music symbols|halfnote}}&lt;br/&gt;''or'' {{tlc|music symbols|half}}&lt;br/&gt;''or'' {{tlc|music symbols|minim}} || {{music symbols|half}}
|-
! Quarter&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15f;&lt;/code&gt; || &amp;#x1d15f;
| {{tlc|music symbols|quarternote}}&lt;br/&gt;''or'' {{tlc|music symbols|quarter}}&lt;br/&gt;''or'' {{tlc|music symbols|crotchet}} || {{music symbols|quarter}}
|-
! Eighth&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d160;&lt;/code&gt; || &amp;#x1d160;
| {{tlc|music symbols|eighthnote}}&lt;br/&gt;''or'' {{tlc|music symbols|eighth}}&lt;br/&gt;''or'' {{tlc|music symbols|quaver}} || {{music symbols|eighth}}
|-
! Beamed&lt;br/&gt;eighth&lt;br/&gt;notes
| ||
| {{tlc|music symbols|eighthnotebeam}}&lt;br/&gt;''or'' {{tlc|music symbols|eighthbeam}}&lt;br/&gt;''or'' {{tlc|music symbols|quaverbeam}} || {{music symbols|eighthbeam}}
|-
! Sixteenth&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d161;&lt;/code&gt; || &amp;#x1d161;
| {{tlc|music symbols|sixteenthnote}}&lt;br/&gt;''or'' {{tlc|music symbols|sixteenth}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaver}} || {{music symbols|sixteenth}}
|-
! Beamed&lt;br/&gt;sixteenth&lt;br/&gt;notes
| ||
| {{tlc|music symbols|sixteenthnotebeam}}&lt;br/&gt;''or'' {{tlc|music symbols|sixteenthbeam}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaverbeam}} || {{music symbols|sixteenthbeam}}
|-
! Thirty-&lt;br/&gt;second&lt;br/&gt;note
| ||
| {{tlc|music symbols|thirty-secondnote}}&lt;br/&gt;''or'' {{tlc|music symbols|thirty-second}}&lt;br/&gt;''or'' {{tlc|music symbols|demisemiquaver}} || {{music symbols|thirty-second}}
|-
! Dot
| . || .
| {{tlc|music symbols|dot}} || {{music symbols|dot}}
|-
! Dotted quarter&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d161;&lt;/code&gt;. || &amp;#x1d161;.
| {{tlc|music symbols|dottedquarter}}&lt;br/&gt;''or'' {{tlc|music symbols|dottedcrotchet}} || {{music symbols|dottedquarter}}
|-
! Dotted half&lt;br/&gt;note
| &lt;code&gt;&amp;amp;#x1d15e;&lt;/code&gt;. || &amp;#x1d15e;.
| {{tlc|music symbols|dottedhalf}}&lt;br/&gt;''or'' {{tlc|music symbols|dottedminim}} || {{music symbols|dottedhalf}}
|-
! Whole&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13b;&lt;/code&gt; || &amp;#x1d13b;
| {{tlc|music symbols|wholerest}}&lt;br/&gt;''or'' {{tlc|music symbols|semibreverest}} || {{music symbols|wholerest}}
|-
! Half&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13c;&lt;/code&gt; || &amp;#x1d13c;
| {{tlc|music symbols|halfrest}}&lt;br/&gt;''or'' {{tlc|music symbols|minimrest}} || {{music symbols|halfrest}}
|-
! Quarter&lt;br/&gt;rest bombaclat

| &lt;code&gt;&amp;amp;#x1d13d;&lt;/code&gt; || &amp;#x1d13d;
| {{tlc|music symbols|quarterrest}}&lt;br/&gt;''or'' {{tlc|music symbols|crotchetrest}} || {{music symbols|quarterrest}}
|-
! Eighth&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13e;&lt;/code&gt; || &amp;#x1d13e;
| {{tlc|music symbols|eighthrest}}&lt;br/&gt;''or'' {{tlc|music symbols|quaverrest}} || {{music symbols|eighthrest}}
|-
! Sixteenth&lt;br/&gt;rest
| &lt;code&gt;&amp;amp;#x1d13f;&lt;/code&gt; || &amp;#x1d13f;
| {{tlc|music symbols|sixteenthrest}}&lt;br/&gt;''or'' {{tlc|music symbols|semiquaverrest}} || {{music symbols|sixteenthrest}}
|-
! Thirty-&lt;br/&gt;second&lt;br/&gt;rest
| ||
| {{tlc|music symbols|thirtysecondrest}}&lt;br/&gt;''or'' {{tlc|music symbols|demisemiquaverrest}} || {{music symbols|thirtysecondrest}}
|}

Some browsers and typefaces support &lt;code&gt;&amp;amp;#x2669;&lt;/code&gt; (&amp;#x2669;) and &lt;code&gt;&amp;amp;#x266a;&lt;/code&gt; (&amp;#x266a;) for quarter and eighth notes, as well as &lt;code&gt;&amp;amp;#x266b;&lt;/code&gt; (&amp;#x266b;) and &lt;code&gt;&amp;amp;#x266c;&lt;/code&gt; (&amp;#x266c;) for beamed eighth-note and sixteenth-note pairs respectively, but since the display of these characters does not match any of the other (non-supported) notes and rests, this template does not use these characters.

;Sample text
In place of the single whole note ({{music symbols|semibreve}}), Chopin writes {{music symbols|eighthrest}} {{music symbols|halfnote}} {{music symbols|quarter}} {{music symbols|eighth}}, completely changing the profile of the music.

==Clefs==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Treble clef&lt;br /&gt;''or'' G-clef
| {{tlc|music symbols|treble}}&lt;br/&gt;''or'' {{tlc|music symbols|trebleclef}}&lt;br/&gt;''or'' {{tlc|music symbols|gclef}} || {{music symbols|trebleclef}}
|-
! Alto&lt;br/&gt;clef
| {{tlc|music symbols|alto}}&lt;br/&gt;''or'' {{tlc|music symbols|altoclef}} || {{music symbols|altoclef}}
|-
! Tenor&lt;br/&gt;clef
| {{tlc|music symbols|tenor}}&lt;br/&gt;''or'' {{tlc|music symbols|tenorclef}} || {{music symbols|tenorclef}}
|-
! C-clef
| {{tlc|music symbols|cclef}}  || {{music symbols|cclef}}
|-
! Bass clef&lt;br /&gt;''or'' F-clef
| {{tlc|music symbols|bass}}&lt;br/&gt;''or'' {{tlc|music symbols|bassclef}}&lt;br /&gt;''or'' {{tlc|music symbols|fclef}}  || {{music symbols|bassclef}}
|-
! Neutral clef
| {{tlc|music symbols|neutral}}&lt;br/&gt;''or'' {{tlc|music symbols|neutralclef}}&lt;br /&gt; || {{music symbols|neutralclef}}
|}

Note that there is no graphical distinction between treble clef and G-clef; alto clef, tenor clef and C-clef; bass clef and F-clef. The names preserve a difference in meaning and make the caption text (for screen readers) different.

== Time signatures ==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Symbol
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! common-time
| {{tlc|music symbols|common-time}} || {{music symbols|common-time}}
|-
! cut-time
| {{tlc|music symbols|cut-time}}&lt;br/&gt;''or'' {{tlc|music symbols|alla-breve}} || {{music symbols|cut-time}}
|-
! 2/4
| {{tlc|music symbols|time|2|4}} || {{music symbols|time|2|4}}
|-
! 3/4
| {{tlc|music symbols|time|3|4}} || {{music symbols|time|3|4}}
|-
! 6/8
| {{tlc|music symbols|time|6|8}} || {{music symbols|time|6|8}}
|-
! 9/8
| {{tlc|music symbols|time|9|8}} || {{music symbols|time|9|8}}
|-
! 12/8
| {{tlc|music symbols|time|12|8}} || {{music symbols|time|12|8}}
|}
For a ''general'' time signature, use &lt;nowiki&gt;{{music symbols|time|&lt;top number&gt;|&lt;bottom number&gt;}}&lt;/nowiki&gt;. This makes use of [[Template:Time signature]], which should not be used on its own.

==Scale degrees==
Scale degrees are often represented as Arabic numerals with a hat on them and thus the root of a scale is &lt;sub&gt;{{music symbols|scale|1}}&lt;/sub&gt;.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:6em;&quot; | Scale degree
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! 1
| {{tlc|music symbols|scale|1}} || {{music symbols|scale|1}}
|-
! 2
| {{tlc|music symbols|scale|2}} || {{music symbols|scale|2}}
|-
! 3
| {{tlc|music symbols|scale|3}} || {{music symbols|scale|3}}
|-
! 4
| {{tlc|music symbols|scale|4}} || {{music symbols|scale|4}}
|-
! 5
| {{tlc|music symbols|scale|5}} || {{music symbols|scale|5}}
|-
! 6
| {{tlc|music symbols|scale|6}} || {{music symbols|scale|6}}
|-
! 7
| {{tlc|music symbols|scale|7}} || {{music symbols|scale|7}}
|-
! 8
| {{tlc|music symbols|scale|8}} || {{music symbols|scale|8}}
|-
! 9
| {{tlc|music symbols|scale|9}} || {{music symbols|scale|9}}
|}

;Sample text
A descending tetrachord could be written as &lt;sub&gt;{{music symbols|scale|4}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|3}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|2}}&lt;/sub&gt;-&lt;sub&gt;{{music symbols|scale|1}}&lt;/sub&gt;.

==Chord symbols==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! style=&quot;width:9em;&quot; | Chord
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! Diminished
| {{tlc|music symbols|dim}}&lt;br/&gt;''or'' {{tlc|music symbols|dimdeg}} || {{music symbols|dim}}&lt;br/&gt;''or'' {{music symbols|dimdeg}}
|-
! Half-diminished
| {{tlc|music symbols|halfdim}}&lt;br/&gt;''or'' {{tlc|music symbols|dimslash}} || {{music symbols|halfdim}}
|-
! Augmented
| {{tlc|music symbols|+}}&lt;br/&gt;''or'' {{tlc|music symbols|aug}} || {{music symbols|aug}}
|-
! Major
| {{tlc|music symbols|delta}}&lt;br/&gt;''or'' {{tlc|music symbols|major}} || {{music symbols|major}}
|}

;Sample text
vii{{music symbols|dim}} becomes vii{{music symbols|dimslash}} in B{{music symbols|flat}} minor by raising the G{{music symbols|b}} to G{{music symbols|natural}}.

III becomes III{{music symbols|aug}} in G{{music symbols|#}} minor by raising the F{{music symbols|#}} to F{{music symbols|x}}.

The F{{music symbols|#}}{{music symbols|major}}&lt;sup&gt;7&lt;/sup&gt; is used to great effect in the last measure of the piece.

==Key signatures==
Key signatures must be typed in using the names of the articles themselves on keys. Note that there is no visual difference between the major and minor key signatures, but the alt text is different.

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
! Sharps/Flats
! style=&quot;width:9em;&quot; | Key signature (major)
! style=&quot;width:16em;&quot; | Template&lt;br/&gt;text 
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
! style=&quot;width:9em;&quot; | Key signature (minor)
! style=&quot;width:16em; | Template&lt;br/&gt;text
! style=&quot;width:6em;&quot; | Template&lt;br/&gt;result
|-
! {{font|color=blue|7{{music symbols|#}}}}
! [[C-sharp major|C{{music symbols|#}} major]]
| {{tlc|music symbols|c-sharp major}} || {{music symbols|c-sharp major}}
! [[A-sharp minor|A{{music symbols|#}} minor]]
| {{tlc|music symbols|a-sharp minor}} || {{music symbols|a-sharp minor}}
|-
! {{font|color=blue|6{{music symbols|#}}}}
! [[F-sharp major|F{{music symbols|#}} major]]
| {{tlc|music symbols|f-sharp major}} || {{music symbols|f-sharp major}}
! [[D-sharp minor|D{{music symbols|#}} minor]]
| {{tlc|music symbols|d-sharp minor}} || {{music symbols|d-sharp minor}}
|-
! {{font|color=blue|5{{music symbols|#}}}}
! [[B major]]
| {{tlc|music symbols|b major}} || {{music symbols|b major}}
! [[G-sharp minor|G{{music symbols|#}} minor]]
| {{tlc|music symbols|g-sharp minor}} || {{music symbols|g-sharp minor}}
|-
! {{font|color=blue|4{{music symbols|#}}}}
! [[E major]]
| {{tlc|music symbols|e major}} || {{music symbols|e major}}
! [[C-sharp minor|C{{music symbols|#}} minor]]
| {{tlc|music symbols|c-sharp minor}} || {{music symbols|c-sharp minor}}
|-
! {{font|color=blue|3{{music symbols|#}}}}
! [[A major]]
| {{tlc|music symbols|a major}} || {{music symbols|a major}}
! [[F-sharp minor|F{{music symbols|#}} minor]]
| {{tlc|music symbols|f-sharp minor}} || {{music symbols|f-sharp minor}}
|-
! {{font|color=blue|2{{music symbols|#}}}}
! [[D major]]
| {{tlc|music symbols|d major}} || {{music symbols|d major}}
! [[B minor]]
| {{tlc|music symbols|b minor}} || {{music symbols|b minor}}
|-
! {{font|color=blue|1{{music symbols|#}}}}
! [[G major]]
| {{tlc|music symbols|g major}} || {{music symbols|g major}}
! [[E minor]]
| {{tlc|music symbols|e minor}} || {{music symbols|e minor}}
|-
! {{font|color=green|0}}
! [[C major]]
| {{tlc|music symbols|c major}} || {{music symbols|c major}}
! [[A minor]]
| {{tlc|music symbols|a minor}} || {{music symbols|a minor}}
|-
! {{font|color=red|1{{music symbols|b}}}}
! [[F major]]
| {{tlc|music symbols|f major}} || {{music symbols|f major}}
! [[D minor]]
| {{tlc|music symbols|d minor}} || {{music symbols|d minor}}
|-
! {{font|color=red|2{{music symbols|b}}}}
! [[B-flat major|B{{music symbols|b}} major]]
| {{tlc|music symbols|b-flat major}} || {{music symbols|b-flat major}}
! [[G minor]]
| {{tlc|music symbols|g minor}} || {{music symbols|g minor}}
|-
! {{font|color=red|3{{music symbols|b}}}}
! [[E-flat major|E{{music symbols|b}} major]]
| {{tlc|music symbols|e-flat major}} || {{music symbols|e-flat major}}
! [[C minor]]
| {{tlc|music symbols|c minor}} || {{music symbols|c minor}}
|-
! {{font|color=red|4{{music symbols|b}}}}
! [[A-flat major|A{{music symbols|b}} major]]
| {{tlc|music symbols|a-flat major}} || {{music symbols|a-flat major}}
! [[F minor]]
| {{tlc|music symbols|f minor}} || {{music symbols|f minor}}
|-
! {{font|color=red|5{{music symbols|b}}}}
! [[D-flat major|D{{music symbols|b}} major]]
| {{tlc|music symbols|d-flat major}} || {{music symbols|d-flat major}}
! [[B-flat minor|B{{music symbols|b}} minor]]
| {{tlc|music symbols|b-flat minor}} || {{music symbols|b-flat minor}}
|-
! {{font|color=red|6{{music symbols|b}}}}
! [[G-flat major|G{{music symbols|b}} major]]
| {{tlc|music symbols|g-flat major}} || {{music symbols|g-flat major}}
! [[E-flat minor|E{{music symbols|b}} minor]]
| {{tlc|music symbols|e-flat minor}} || {{music symbols|e-flat minor}}
|-
! {{font|color=red|7{{music symbols|b}}}}
! [[C-flat major|C{{music symbols|b}} major]]
| {{tlc|music symbols|c-flat major}} || {{music symbols|c-flat major}}
! [[A-flat minor|A{{music symbols|b}} minor]]
| {{tlc|music symbols|a-flat minor}} || {{music symbols|a-flat minor}}
|}

&lt;includeonly&gt;
[[Category:Music templates|{{BASEPAGENAME}}]]

&lt;/includeonly&gt;</text>
      <sha1>log3gpx8zpk13s72cg1iwgcmdc66uy8</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Jeanette</title>
    <ns>3</ns>
    <id>115712</id>
    <revision>
      <id>2624765</id>
      <parentid>2194400</parentid>
      <timestamp>2024-05-02T16:57:23Z</timestamp>
      <contributor>
        <ip>41.122.132.25</ip>
      </contributor>
      <comment>/* submitted book */ Reply</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7143" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello Jeanette, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en &lt;kbd&gt;#wikiversity-en&lt;/kbd&gt;].
* Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity).
&lt;/div&gt;

&lt;br clear=&quot;both&quot;/&gt;

You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 01:26, 3 July 2011 (UTC)&lt;/div&gt;
{{Robelbox/close}}
{{Robelbox|title=[[Wikiversity:Mascot contest|Mascot welcome]]|theme=5|icon=Crystal Clear app gnome.png|iconwidth=48px}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
[[Image:Jack-Russell-Terrier.jpg|48px|left]]
'''Woof!''' My name is [[User:Jack Russell|Jack Russell]]. I am a dog and a [[Wikiversity:Mascot contest|Wikiversity mascot]]. I am pretty new around [[Wikiversity]]. Perhaps we can learn together! ''Tail wag''...
&lt;/div&gt;
{{Robelbox/close}}
==Welcome to [[Motivation and emotion]]==
Hi {{PAGENAME}}. Welcome to the unit and to [[Wikiversity]]! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:54, 5 August 2011 (UTC)

== Where personal comments for users are best placed ==

[http://en.wikiversity.org/w/index.php?title=User:Wikitwit&amp;diff=773782&amp;oldid=767326]. Generally, personal communications with a user should be placed on the user's talk page, not on the user page. Mostly, with a few exceptions, only users are allowed to edit their own user page. Not a problem, I moved your comment to the User's talk page (and users often have their preferences set to send them an email when their user page is edited, so it's much better to place a comment there). Just letting you know for future reference. Thanks for participating at Wikiversity, it's a pleasure to serve you. --[[User:Abd|Abd]] 19:49, 11 September 2011 (UTC)

==Username==
The issue is a bit complicated. One option is to do nothing and have different logins on Wikiversity and Wiki Commons. This is the easiest solution, but its the messiest. Another option is to find a new user name that could be unified across all the Wikimedia Foundation projects. This is a neater solution and generally better. We would rename your current Jeanette account which will bring all your edit history etc. into the new account name. That account name would then work across all the WMF projects. It is not a big deal to do and then you are set up for life with a unified WMF account. If we're going to rename, then probably better to do sooner rather than later. Your old user page would be redirected automatically to your new user page. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 20 October 2011 (UTC)
: On closer look, actually, maybe you could claim the Jeanette account and have it unified - it isn't actively used elsewhere: http://toolserver.org/~quentinv57/tools/sulinfo.php?username=Jeanette but someone else registered it on other projects at some point earlier. Technically, this would be to apply to usurp the user name on the other projects because they are inactive. This can take a bit of time and admin to sort out - but we'd start with trying to usurp Jeanette on Wiki Commons. Depends a bit how far into wiki-land you want to go... {{smile}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:47, 20 October 2011 (UTC)
:: Place a request here [[Commons:Commons:Changing username/Usurp requests]] to usurp User:Jeanette and see what they say. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:58, 20 October 2011 (UTC)
::: I fixed the link above - external links would be http://commons.wikimedia.org/wiki/Commons:Changing_username/Usurp_requests -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:03, 20 October 2011 (UTC)

==Table==
I'm not sure why that [[shame]] table was misbehaving - but I fiddled with it and hopefully it's in the right spot now and the bottom row is how you wanted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:04, 27 October 2011 (UTC)

== submitted book ==

Can't write anymore. Counted words - 4449 so should be under according to wordcount. [[User:Jeanette|Jeanette]] 20:47, 6 November 2011 (UTC)
Make that 4458 words. [[User:Jeanette|Jeanette]] 20:51, 6 November 2011 (UTC)
: I had a quick look and made some minor 'wikifications' -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:11, 6 November 2011 (UTC)
::okat [[Special:Contributions/41.122.132.25|41.122.132.25]] ([[User talk:41.122.132.25|discuss]]) 16:57, 2 May 2024 (UTC)</text>
      <sha1>67fjbxga82mnsfta57nvb6tzl41xv1b</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Marshallsumter</title>
    <ns>3</ns>
    <id>117349</id>
    <revision>
      <id>2624859</id>
      <parentid>2603091</parentid>
      <timestamp>2024-05-02T23:10:33Z</timestamp>
      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* Reminder to vote now to select members of the first U4C */ new section</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18479" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello Marshallsumter, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Signature icon.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;

&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en &lt;kbd&gt;#wikiversity-en&lt;/kbd&gt;].
* Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity).
&lt;/div&gt;

&lt;br clear=&quot;both&quot;/&gt;

You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Abd|Abd]] 04:25, 24 August 2011 (UTC)&lt;/div&gt;
{{Robelbox/close}}

== Most Active Wikiversity User for January 2013 ==

{| style=&quot;border: 1px solid gray; background-color: #ffffff;&quot;
|rowspan=&quot;2&quot; valign=&quot;middle&quot; | [[Image:Learningcycle.png|100px]]
|rowspan=&quot;2&quot; |
|style=&quot;font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;&quot; | '''The Learning Cycle Barnstar'''
|-
|style=&quot;vertical-align: middle; border-top: 1px solid gray;&quot; |  Most Active Wikiversity User for January 2013
|}

Marshallsumter, I was reviewing the list of active users for this past month and noticed you had by far the most edits in January.  Keep up the good work!  -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:04, 1 February 2013 (UTC)

== Barnstar for you! ==

{| style=&quot;border: 1px solid gray; background-color: #ffffff;&quot;
|rowspan=&quot;2&quot; valign=&quot;middle&quot; | [[Image:Star constellation.png|100px]]
|rowspan=&quot;2&quot; |
|style=&quot;font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;&quot; | '''The astronomy barnstar'''
|-
|style=&quot;vertical-align: middle; border-top: 1px solid gray;&quot; |  Thank you for the massive edits on astronomy! [[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]|[[Special:Contributions/Goldenburg111|contribs]]) 18:49, 25 December 2013 (UTC)
|}
{| style=&quot;border: 1px solid gray; background-color: #ffffff;&quot;
|rowspan=&quot;2&quot; valign=&quot;middle&quot; | [[Image:Original_Barnstar.png|100px]]
|rowspan=&quot;2&quot; |
|style=&quot;font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;&quot; | '''The Original Barnstar'''
|-
|style=&quot;vertical-align: middle; border-top: 1px solid gray;&quot; |  Thank you for your help with [[Research in programming Wikidata]]! -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 05:45, 30 May 2017 (UTC)
|}

==See also==
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{{clear}}

==Recent contributions from WikiJournal of Science Editorial Board==

{| class=&quot;wikitable&quot; style=&quot;margin:auto;&quot;
 ! rowspan=2 | Editor/Associate Editor
 ! rowspan=2 | Date approved 
 ! rowspan=2 | Username
 ! colspan=&quot;2&quot; |Journal contributions 
 ! rowspan=2 | Current status 
 |-
 !Earliest contribution !! Latest contribution 
 |-
 !1. Editor: Guy Vandegrift 
 | 18 January 2016‎ || [[Special:Contributions/Guy_vandegrift|Guy vandegrift]] || 18 January 2016 || 17 November 2019‎ || Honorary 
 |-
 !2. Editor: Michael L. Umbricht 
 | 18 January 2016‎ || [[Special:Contributions/Mu301|Michael L. Umbricht]] || 18 January 2016‎ || 25 March 2020 || Associate editor, Inactive 
 |-
 !3. Advisor: Mikael Häggström 
 | 21 January 2016‎ || [[Special:Contributions/Mikael_Häggström|Mikael Häggström]] || 21 January 2016‎ || 21 August 2022 || Withdrawn 
 |-
 !4. former Editor-in-Chief: Felipe Schenone 
 | 12 December 2016‎ || [[Special:Contributions/Sophivorus|Sophivorus]] || 12 December 2016‎ || 31 May 2020‎ || Withdrawn 
 |-
 !5. Editor: Henry Hoff 
 | 5 November 2017 || [[Special:Contributions/Marshallsumter|Marshallsumter]] || 10 January 2017‎ || 1 February 2024 || Active
 |-
 !6. former Editor-in-Chief: Thomas Shafee
 | 30 October 2017 || [[Special:Contributions/Evolution_and_evolvability|Evolution and evolvability]] || 13 October 2017‎ || 19 September 2023 || Active 
 |-
 !7. Editor: W. Brian Whalley 
 | 5 November 2017‎ || [[Special:Contributions/W.BrianWhalley|W.BrianWhalley]] || 19 October 2017‎ || 18 November 2018 || Inactive 
 |-
 !8. Editor: Markus Pössel 
 | 5 November 2017 || [[Special:Contributions/Markus_Pössel|Markus Pössel]] || 20 October 2017 || 22 March 2020 || Withdrawn 
 |-
 !9. Editor: Ian Alexander 
 | 5 November 2017 || [[Special:Contributions/Chiswick_Chap|Chiswick Chap]] || 21 October 2017 || 26 June 2023 || Active 
 |-
 !10. Editor: Joanna Argasinska 
 | 5 November 2017 || [[Special:Contributions/Joanna_Argasinska|Joanna Argasinska]] || 24 October 2017 || 26 February 2019 || Associate editor, Inactive 
 |-
 !11. Editor: Florian Weller 
 | 24 November 2017 || [[Special:Contributions/Elmidae|Elmidae]] || 30 October 2017 || 17 December 2020 || Withdrawn 
 |-
 !12. Editor: Marc Robinson-Rechavi 
 | 24 November 2017 || [[Special:Contributions/Marcrr|Marcrr]] || 2 November 2017 || 22 September 2022 || Withdrawn 
 |-
 !13. Editor: Daniele Pugliesi 
 | 24 November 2017 || [[Special:Contributions/Daniele_Pugliesi|Daniele Pugliesi]] || 5 November 2017 || 11 June 2018 || Withdrawn 
 |-
 !14. Editor: Sylvain Ribault 
 | 24 November 2017 || [[Special:Contributions/Sylvain_Ribault|Sylvain Ribault]] || 6 November 2017 || 30 December 2023 || Inactive 
 |-
 !15. Editor: Melanie Stefan 
 | 24 November 2017 || [[Special:Contributions/Mstefan|Mstefan]] || 7 November 2017 || 13 September 2023 || Active 
 |-
 !16. Editor: Jack Nunn 
 | 24 November 2017 || [[Special:Contributions/Jacknunn|Jacknunn]] || 9 November 2017 || 31 January 2024 || Active 
 |-
 !17. Editor: Sridhar Gutam 
 | 24 November 2017 || [[Special:Contributions/Gutam2000|Gutam2000]] || 13 November 2017 || 13 November 2017 || Inactive 
 |-
 !18. Editor: Shampa Ghosh 
 | 30 November 2017 || [[Special:Contributions/Shampa.ghosh|Shampa.ghosh]] || 25 November 2017 || 10 December 2017 || Withdrawn 
 |-
 !19. Editor: Jitendra Kumar Sinha 
 | 30 November 2017 || [[Special:Contributions/G10sinha|G10sinha]] || 25 November 2017 || 12 September 2022 || Withdrawn 
 |-
 !20. Editor: Thijs van Vlijmen 
 | 6 March 2018 || [[Special:Contributions/Van_Vlijmen|Van Vlijmen]] || 30 November 2017 || 7 March 2018 || Inactive 
 |-
 !21. Editor: Roger Watson 
 | 16 January 2018 || [[Special:Contributions/Parveenali|Roger Watson]] || 11 January 2018 || 11 January 2018 || Withdrawn 
 |-
 !22. Editor: Jack Brooks 
 | 15 April 2018 || [[Special:Contributions/JackBrooksDr|Jack Brooks]] || 15 March 2018 || 4 June 2018 || Withdrawn 
 |-
 !23. Editor: Kelee Pacion 
 | 21 April 2018 || [[Special:Contributions/Saguaromelee|Kelee Pacion]] || 29 March 2018 || 9 August 2021 || Inactive 
 |-
 !24. Editor: Edmund F. Palermo 
 | 21 April 2018 || [[Special:Contributions/EdPalermoRPI|EdPalermoRPI]] || 20 March 2018 || 22 November 2020 || Inactive 
 |-
 !25. Editor: Tina Qin 
 | 21 April 2018 || [[Special:Contributions/VandyChem5600|VandyChem5600]] || 30 March 2018 || 1 June 2018 || Inactive 
 |-
 !26. Editor: Loren Cobb 
 | 21 April 2018 || [[Special:Contributions/Aetheling|Aetheling]] || 30 March 2018 || 30 March 2018 || Inactive 
 |-
 !27. Editor: Paula Diaconescu 
 | 21 April 2018 || [[Special:Contributions/Pauladiaconescu|Pauladiaconescu]] || 31 March 2018 || 31 March 2018 || Inactive 
 |-
 !28. Editor-in-Chief: Andrew Leung 
 | 21 April 2018 || [[Special:Contributions/OhanaUnited|OhanaUnited]] || 31 March 2018 || 22 January 2024 || Active 
 |-
 !29. Editor: José Lages
 | 21 April 2018 || [[Special:Contributions/Joselages|Joselages]] || 31 March 2018 || 4 June 2018 || Inactive 
 |-
 !30. Editor: Muhammad Elhossary 
 | 16 April 2018 || [[Special:Contributions/Muhammad_elhossary|Muhammad elhossary]] || 4 April 2018 || 16 April 2018 || Withdrawn 
 |-
 !31. Editor: Thais C. Morata
 | 20 May 2018 || [[Special:Contributions/TMorata|TMorata]] || 5 April 2018 || 15 January 2024 || Active 
 |-
 !32. Editor: Konrad U. Förstner
 | 20 May 2018 || [[Special:Contributions/Konrad_Foerstner|Konrad Foerstner]] || 15 April 2018 || 10 December 2018 || Inactive 
 |-
 !33. Editor: Jonathan Holland
 | 20 May 2018 || [[Special:Contributions/Ensahequ|Ensahequ]] || 3 June 2018 || 13 June 2020 || Inactive 
 |-
 !34. Editor: Vinod Scaria 
 | 20 May 2018 || [[Special:Contributions/Sdoniv|Sdoniv]] || 19 June 2018 || 5 October 2018 || Inactive 
 |-
 !35. Editor: Hemachander Subramanian 
 | 15 November 2018 || [[Special:Contributions/HemachanderTBio|HemachanderTBio]] || 19 June 2018 || 10 February 2020 || Associate editor, Inactive 
 |-
 !36. Editor: Ayush Bhardwaj 
 | Declined || [[Special:Contributions/Ayushb15|Ayush Bhardwaj]] || 5 November 2018 || 7 June 2019 || Withdrawn 
 |-
 !37. Editor: Gorla Praveen 
 | Declined || [[Special:Contributions/Gorlapraveen123|Gorlapraveen123]] || 23 November 2018 || 8 August 2022 || Associate editor, Inactive 
 |-
 !38. Editor: Ed Baker 
 | 31 January 2019 || [[Special:Contributions/Edwbaker|Edwbaker]] || 5 December 2018 || 19 August 2019 || Inactive 
 |-
 !39. Editor: Karthik Muthineni 
 | Declined || [[Special:Contributions/Muthineni|Karthik Muthineni]] || 11 February 2019 || 11 February 2019 || Withdrawn 
 |-
 !40. Editor: David Wirth 
 | Declined || [[Special:Contributions/Dwirth9|David Wirth]] || 28 May 2019 || 17 June 2019 || Withdrawn 
 |-
 !41. Editor: Scott A Thomson 
 | 19 June 2019 || [[Special:Contributions/Faendalimas|Faendalimas]] || 10 June 2019 || 30 November 2023 || Active 
 |-
 !42. Editor: Dan Graur 
 | 29 August 2019 || [[Special:Contributions/Dogrt|Dogrt]] || 2 July 2019 || 26 August 2019 || Associate editor, Inactive 
 |-
 !43. Editor: Elizabeth Van Volkenburgh 
 | 29 August 2019 || [[Special:Contributions/2601:602:8A01:4153:E169:417E:2F9D:9436|Elizabeth Van Volkenburgh]] || 2 August 2019 || 2 August 2019 || Associate editor, Inactive 
 |-
 !44. Editor: Tony Ross-Hellauer 
 | 24 November 2019 || [[Special:Contributions/Tonyross79|Tonyross79]] || 20 September 2019 || 15 October 2019 || Associate editor, Inactive 
 |-
 !45. Editor: MGH Zaidi 
 | Declined || [[Special:Contributions/Dr.MGH_Zaidi|MGH Zaidi]] || 14 October 2019 || 14 October 2019 || Withdrawn 
 |-
 !46. Mad Ball Price 
 | 25 November 2019 || [[Special:Contributions/Mad_Price_Ball|Mad Price Ball]] || 21 November 2019 || 2 January 2020 || Inactive 
 |-
 !47. Editor: Jeff Lundeen 
 | 28 January 2020 || [[Special:Contributions/J_S_Lundeen|J S Lundeen]] || 16 December 2019 || 24 October 2021 || Inactive 
|-
 !48. Editor: Rosemary J Redfield 
 | 3 May 2020 || [[Special:Contributions/Rosieredfield|Rosieredfield]] || 30 January 2020 || 12 September 2022 || Withdrawn 
 |-
 !49. Editor: Yulia Sevryugina 
 | 7 August 2020 || [[Special:Contributions/MLibrarian|MLibrarian]] || 17 February 2020 || 29 January 2021 || Inactive 
 |-
 !50. Editor: Emanuele Natale 
 | 11 October 2020 || [[Special:Contributions/Natematic|Natematic]] || 8 April 2020 || 18 May 2023 || Active 
 |-
 !51. Editor: Moritz Schubotz 
 | 7 August 2020 || [[Special:Contributions/Physikerwelt|Physikerwelt]] || 3 July 2020 || 21 July 2023 || Active 
 |-
 !52. Editor: Jong Bhak 
 | 3 December 2020 || [[Special:Contributions/Jongbhak|Jongbhak]] || 28 October 2020 || 16 December 2020 || Inactive 
 |-
 !53. Editor: Fernando Pinheiro Andutta 
 | Timed out || [[Special:Contributions/49.182.51.145|Fernando Pinheiro Andutta]] || 2 November 2020 || 2 November 2020 || Withdrawn 
 |-
 !54. Editor: Roger M. Rosewall 
 | Declined || [[Special:Contributions/Rosewall2020|Rosewall2020]] || 6 January 2021 || 6 January 2021 || Inactive 
 |-
 !55. Editor: Michel Bakni 
 | 19 November 2022 || [[Special:Contributions/Michel_Bakni|Michel Bakni]] || 4 April 2021 || 2 November 2023 || Active 
 |-
 !56. Editor: Daniel Gliksman 
 | 22 November 2022 || [[Special:Contributions/2A02:810A:8CC0:54D0:F4A1:6610:E613:2CBE|Lucidan]] || 14 September 2022 || 6 October 2022 || Associate editor, Inactive 
 |-
 !57. Editor: Alex O. Holcombe 
 | 22 November 2022 || [[Special:Contributions/Aoholcombe|Aoholcombe]] || 11 January 2024 || 5 December 2023 || Associate editor, Active 
 |-
 !58. Editor: Kevin Moerman 
 | 19 June 2023 || [[Special:Contributions/KevinMoerman|KevinMoerman]] || 27 April 2023 || 26 April 2023 || Active 
 |-
 !59. Editor: Bala Zoology 
 | Withdrawn || [[Special:Contributions/Bala_Zoology|Solamuthu Balamurugan]] || 20 June 2023 || 16 June 2023 || Withdrawn 
 |-
 !59. Editor: Mariselvam 
 | Not responsive || [[Special:Contributions/Maiselvam88|Maiselvam]] || 30 June 2023 || 30 June 2023‎ || Withdrawn 
 |-
|}
: Thanks for compiling this list. I just want to point out that this is solely based on contributions verifiable on-wiki. It does not take into the account of off-wiki activities (e.g. Kelee Pacion has been arranging for meetings well into May 2022 and Jack Nunn voiced his opinion of an article in the mailing list in August 2022, yet their contributions based from on-wiki history would not reveal this aspect). [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 15:18, 19 October 2022 (UTC)
:: That's a good and valid point! I've only attended one or maybe two meetings away from Wikiversity but I do appreciate the efforts those attending these meetings have been making! I've kept Kelee and Jack as active participants even though there are few Wikiversity edits toward WikiJournals. If there are any I've listed as inactive that also are active such as at these meetings feel free to change them to active. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 01:55, 20 October 2022 (UTC)
::: Updated for recent editors and results. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 17:47, 12 June 2023 (UTC)
:::: Updated for recent editors and results. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 18:46, 1 February 2024 (UTC)

==Continental shelf inhabitants during the Last Glacial Maximum==

::::I am starting to write my &quot;Adventure of the Atlantis Hypothesis&quot;.
::::I am using a lot of information about the Atlantis  Hyopothesis. I am cheating with current events and lots of things. I need YOUR advice. 
::::Can you email me at jgarner812 at gmail dot com?
::::I lost your email 
::::[[User:RAYLEIGH22|RAYLEIGH22]] ([[User talk:RAYLEIGH22|discuss]] • [[Special:Contributions/RAYLEIGH22|contribs]]) 01:24, 4 November 2023 (UTC)
:::::Emailed today. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 04:52, 16 December 2023 (UTC)

== Reminder to vote now to select members of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – vote reminder|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – vote reminder}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear Wikimedian,

You are receiving this message because you previously participated in the UCoC process.

This is a reminder that the voting period for the Universal Code of Conduct Coordinating Committee (U4C) ends on May 9, 2024. Read the information on the [[m:Universal Code of Conduct/Coordinating Committee/Election/2024|voting page on Meta-wiki]] to learn more about voting and voter eligibility.

The Universal Code of Conduct Coordinating Committee (U4C) is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community members were invited to submit their applications for the U4C. For more information and the responsibilities of the U4C, please [[m:Universal Code of Conduct/Coordinating Committee/Charter|review the U4C Charter]].

Please share this message with members of your community so they can participate as well.

On behalf of the UCoC project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 23:10, 2 May 2024 (UTC)
&lt;!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Universal_Code_of_Conduct/Coordinating_Committee/Election/2024/Previous_voters_list_2&amp;oldid=26721207 --&gt;</text>
      <sha1>r3f1qwz0hrrg3skv9w959lecl3nz781</sha1>
    </revision>
  </page>
  <page>
    <title>Mathematics/Astronomy</title>
    <ns>0</ns>
    <id>124037</id>
    <revision>
      <id>2624936</id>
      <parentid>2619178</parentid>
      <timestamp>2024-05-03T06:59:50Z</timestamp>
      <contributor>
        <ip>204.219.240.33</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="85716" xml:space="preserve">[[Image:Superclusters atlasoftheuniverse.gif|thumb|250px|right|Our universe within 1 billion light-years (307 Mpc) of Earth is shown to contain the local [[w:supercluster|supercluster]]s, [[w:galaxy filament|galaxy filament]]s and voids. Credit: Richard Powell.]]
Although most of the mathematics needed to understand the information acquired through astronomical observation comes from physics, there are special needs from situations that intertwine mathematics with phenomena that may not yet have sufficient physics to explain the observations. These two uses of mathematics make '''mathematical astronomy''' a continuing challenge.

Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera, specifically its charge-coupled device (CCD) detector, that is attached to the telescope basically converts or counts photons or electrons and records a series of numbers (the counts) - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.

An initial use of mathematics in astronomy is counting [[Radiation astronomy/Entities|entities]], sources, or objects in the sky.

Objects may be counted during the daytime or night.

One use of mathematics is the calculation of distance to an object in the sky.

{{clear}}

==Notations==

'''Notation''': let the symbol &lt;math&gt;\oplus&lt;/math&gt; indicate the [[Earth]].

'''Notation''': let the symbol '''ʘ''' or &lt;math&gt;\odot&lt;/math&gt; indicate the [[Stars/Sun|Sun]].

'''Notation''': let the symbol &lt;math&gt; I_{\odot} &lt;/math&gt; indicate the total solar irradiance.

'''Notation''': let the symbol &lt;math&gt; L_V &lt;/math&gt; indicate the solar visible luminosity.

'''Notation''': let the symbol &lt;math&gt; L_{\odot} &lt;/math&gt; indicate the solar bolometric luminosity.

'''Notation''': let the symbol &lt;math&gt; L_{bol} &lt;/math&gt; indicate the solar bolometric luminosity.

'''Notation''': let the symbol &lt;math&gt;M_{bol}&lt;/math&gt; represent the '''bolometric magnitude''', the total energy output.

'''Notation''': let the symbol &lt;math&gt;M_V&lt;/math&gt; represent the '''visual magnitude'''.

'''Notation''': let the symbol &lt;math&gt; M_{\odot} &lt;/math&gt; indicate the solar mass.

'''Notation''': let the symbol &lt;math&gt; Q_{\odot} &lt;/math&gt; represent the net solar charge.

'''Notation''': let the symbol &lt;math&gt;R_\oplus&lt;/math&gt; indicate the Earth's radius.

'''Notation''': let the symbol &lt;math&gt;R_J&lt;/math&gt; indicate the radius of Jupiter.

'''Notation''': let the symbol &lt;math&gt;R_{\odot}&lt;/math&gt; indicate the solar radius.

:{| border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;none&quot;
|-
! colspan=&quot;3&quot; |'''Notational locations'''
|-
|'''Weight'''
|'''Oversymbol'''
|'''Exponent'''
|-
|'''Coefficient'''
|'''Variable'''
|'''Operation'''
|-
|'''Number'''
|'''Range'''
|'''Index'''
|-
|}

For each of the notational locations around the central '''Variable''', conventions are often set by consensus as to use. For example, '''Exponent''' is often used as an exponent to a number or variable: 2&lt;sup&gt;-2&lt;/sup&gt; or x&lt;sup&gt;2&lt;/sup&gt;.

In the '''Notation'''s at the top of this section, '''Index''' is replaced by symbols for the Sun (ʘ), [[Earth]] (&lt;math&gt;R_\oplus&lt;/math&gt;), or can be for Jupiter (J) such as &lt;math&gt;R_J&lt;/math&gt;.

A common '''Oversymbol''' is one for the average &lt;math&gt;\overline{Variable}&lt;/math&gt;.

'''Operation''' may be replaced by a function, for example.

All notational locations could look something like

:{| border=&quot;0&quot; cellpadding=&quot;3&quot; cellspacing=&quot;0&quot; align=&quot;none&quot;
|-
|bx
|&lt;math&gt;-&lt;/math&gt;
|x = n
|-
|a
|&lt;math&gt;\sum&lt;/math&gt;
|f(x)
|-
|n
|→
|∞
|-
|}
where the center line means &quot;a x Σ f(x)&quot; for all added up values of f(x) when x = n from say 0 to infinity with each term in the sum before summation multiplied by bn, then divided by n for an average whenever n is finite.

==Abstractions==
{{main|Abstractions}}
A '''nomy''' (Latin ''nomia'') is a &quot;system of [[Laws|law]]s governing or [the] sum of [[knowledge]] regarding a (specified) field.&quot;&lt;ref name=Gove&gt;{{ cite book
|author=
|title=Webster's Seventh New Collegiate Dictionary
|publisher=G. &amp; C. Merriam Company
|location=Springfield, Massachusetts
|date=1963
|editor=Philip B. Gove
|pages=1221
|bibcode=
|doi=
|pmid=
|isbn=
}}&lt;/ref&gt; ''Nomology'' is the &quot;[[What is science?|science]] of physical and logical laws.&quot;&lt;ref name=Gove/&gt;

'''Def.''' the quality of dealing with ideas rather than events is called '''abstraction'''.

'''Def.''' the act of the theoretical way of looking at things; something that exists only in idealized form is called '''abstraction'''.

==Relations==
{{main|Relations}}
'''Notation''': let the relation symbol '''≠''' indicate that two expressions are different.

For example, 2 x 3 ≠ 5 x 7.

'''Notation''': let the relation symbol '''~''' represent '''similar to'''.

For example, depending on the scale involved, 7 ~ 8 on a scale of 10, 7/10 = 0.7 and 8/10 = 0.8. relative to numbers between 0.5 and 1.0, 0.7 ~ 0.8, but 0.2 ≁ 0.7.

Similarity may be close such as 0.7 ≈ 0.8, but 0.5 ~ 0.8. Or similarity may include equality, 5 ± 3 ≃ 4 ± 2. When the degree of equality is greater than the degree of similarity, the symbol ≅ is used. The reverse is represented by ≊.

==Differences==
{{main|Abstractions/Differences|Differences}}
Here's a [[Definitions/Theory#Theoretical definition|theoretical definition]]:

'''Def.''' an abstract relation between identity and sameness is called a '''difference'''.

'''Notation''': let the symbol &lt;math&gt;\Delta&lt;/math&gt; represent '''difference in'''.

==Order==
{{main|Order}}
Ordering numbers may mean listing them in increasing value. For example, 2 is less than 3 so that in increasing order 2,3 is the list.

'''Notation''': let the symbol '''&gt;''' represent '''greater than'''.

For example, the integer five (5) is greater than the integer (2): 5 &gt; 2.

'''Notation''': let the symbol '''&lt;''' represent '''less than'''.

For, example, 2 &lt; 3.

==Numbers==
{{main|Numbers}}
:{|style=&quot;border:1px solid #ddd; text-align:center; margin:auto&quot; cellspacing=&quot;20&quot;
|&lt;math&gt;1, 2, 3,\ldots\!&lt;/math&gt; || &lt;math&gt;\ldots,-2, -1, 0, 1, 2\,\ldots\!&lt;/math&gt; || &lt;math&gt; -2, \frac{2}{3}, 1.21\,\!&lt;/math&gt; || &lt;math&gt;-e, \sqrt{2}, 3, \pi\,\!&lt;/math&gt; || &lt;math&gt;2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!&lt;/math&gt;
|-
|[[Natural number]]s|| [[Integer]]s || [[w:Rational number|Rational number]]s || [[Real Numbers|Real number]]s || [[Complex Numbers|Complex number]]s
|}
[[Image:Avogadro's number in e notation.jpg|thumb|upright|200px|A calculator display showing an approximation to the [[w:Avogadro constant|Avogadro constant]] in E notation. Credit: [[commons:User:PRHaney|PRHaney]].]]
'''Scientific notation''' (more commonly known as '''standard form''') is a way of writing numbers that are too big or too small to be conveniently written in decimal form.  Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers.

{| class=&quot;wikitable&quot; style=&quot;float:left&quot;
|-
!Standard decimal notation
!Normalized scientific notation
|-
| 2
|{{val|2|e=0}}
|-
| 300
|{{val|3|e=2}}
|-
| 4,321.768
|{{val|4.321768|e=3}}
|-
| -53,000
|{{val|-5.3|e=4}}
|-
| 6,720,000,000
|{{val|6.72|e=9}}
|-
| 0.2
|{{val|2|e=-1}}
|-
| 0.000&amp;nbsp;000&amp;nbsp;007&amp;nbsp;51
|{{val|7.51|e=-9}}
|}

A '''metric prefix''' or '''SI prefix''' is a [[w:unit prefix|unit prefix]] that precedes a basic unit of measure to indicate a [[w:decimal|decadic]] [[w:multiple (mathematics)|multiple]] or [[w:fraction (mathematics)|fraction]] of the unit. Each prefix has a unique symbol that is prepended to the unit symbol.

{{SI prefixes}}

A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes [[w:Significant figures#Identifying significant digits|indicated to be significant.]]

Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures—1, 2, 3, 0, and 4; the two zeroes serve only as placeholders and add no precision to the original number.

When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but all of the place holding zeroes are incorporated into the exponent. Following these rules, 1,230,400 becomes 1.2304 x 10&lt;sup&gt;6&lt;/sup&gt;.

It is customary in scientific measurements to record all the significant digits from the measurements, for example, 1,230,400, but the measurement may have introduced an error which when calculated indicates the last significant digit has a range of values where the most likely one is the &quot;4&quot;. The range may be 3-5 so that the last significant digit plus this error may be written as (4,1) meaning 4-1=3 and 4+1=5.

Another example of significant digits is the speed of all [[w:massless particle|massless particle]]s and associated [[w:field (physics)|field]]s—including [[w:electromagnetic radiation|electromagnetic radiation]] such as [[w:light|light]]—in vacuum ... [The most accurate value is] 299792.4562±0.0011 [km/s].&lt;ref name=&quot;NIST heterodyne&quot;&gt;{{cite journal
 |last1=Evenson 
 |first1=KM 
 |display-authors=etal
 |year=1972
 |title=Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser
 |journal=Physical Review Letters
 |volume=29
 |issue=19 |pages=1346–49
 |doi=10.1103/PhysRevLett.29.1346
|bibcode=1972PhRvL..29.1346E
}}&lt;/ref&gt; The magnitude of the speed is 299792.4562 and the actual measured variation is ±0.0011 so that the last two significant digits &quot;62&quot; are most likely within a variation from &quot;51&quot; to &quot;73&quot;.

Most [[w:calculator|calculator]]s and many [[w:computer program|computer program]]s present very large and very small results in E notation. The [[w:E|letter ''E'' or ''e'']] is often used to represent ''times ten raised to the power of'' (which would be written as &quot;x&amp;nbsp;10&lt;sup&gt;''b''&lt;/sup&gt;&quot;) [where ''b'' represents a number] and is followed by the value of the exponent.

'''Def.''' any real number that cannot be expressed as a ratio of two integers is called an '''irrational number'''.

'''Def.''' incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof is called '''uncountable'''.

An uncountable set of numbers such as the irrational numbers lies somewhere between a finite set of numbers, for example, the set of natural factors of 6: {1,2,3,6}, and an infinite set of numbers such as the [[w:Natural number|natural number]]s.

'''Def.''' the branch of pure mathematics concerned with the properties of integers is called '''number theory'''.

==Arithmetics==
{{main|Arithmetics}}
Arithmetic involves the manipulation of numbers.

===Theory of arithmetic===

'''Def.''' the [[wikt:mathematics|mathematics]] of [[wikt:number|number]]s ([[wikt:integer|integer]]s, [[wikt:rational number|rational number]]s, [[wikt:real number|real number]]s, or [[wikt:complex number|complex number]]s) under the operations of [[wikt:addition|addition]], [[wikt:subtraction|subtraction]], [[wikt:multiplication|multiplication]], and [[wikt:division|division]] is called an '''arithmetic'''.

'''Def.''' a symbol ( = ) used in mathematics to indicate that two values are the same is called '''equals''', or '''equal to'''.

Consider the integers: 1 and 2. The statement, &quot;1 + 2 = 3&quot;, contains the operation + (addition) and the relation = (equals).

===Exponentials===
{{main|Exponentials}}
The exponential can require a different operator of arithmetic.

The number '''&lt;math&gt;e&lt;/math&gt;''' is an important [[w:mathematical constant|mathematical constant]], approximately equal to 2.71828, that is the base of the [[w:natural logarithm|natural logarithm]].&lt;ref&gt;Oxford English Dictionary, 2nd ed.: [http://oxforddictionaries.com/definition/english/natural%2Blogarithm natural logarithm]&lt;/ref&gt;

:&lt;math&gt;e = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots&lt;/math&gt; 

: e&lt;sup&gt;2&lt;/sup&gt; + e&lt;sup&gt;3&lt;/sup&gt; ≠ e&lt;sup&gt;5&lt;/sup&gt;. Yet
: e&lt;sup&gt;2&lt;/sup&gt; x e&lt;sup&gt;3&lt;/sup&gt; = e&lt;sup&gt;(2 + 3)&lt;/sup&gt; = e&lt;sup&gt;5&lt;/sup&gt;.

===Logarithms===
{{main|Logarithms}}
'''Def.''' For a number [...], the [exponent or power] to which a given ''base'' number must be raised in order to obtain [the number] is called a '''logarithm'''.

Consider 10&lt;sup&gt;3&lt;/sup&gt;, the ''base'' number is 10. The exponent is 3. The number itself is 1000. Using logarithm notation
: &lt;math&gt;\log_{10} 1000 = 3.&lt;/math&gt;

For 2&lt;sup&gt;4&lt;/sup&gt;,
: &lt;math&gt; \log_{2} 16 = 4.&lt;/math&gt;

For e&lt;sup&gt;4&lt;/sup&gt;,
: &lt;math&gt; \ln_{e} e^4 = 4.&lt;/math&gt;

'''Notation''': let the symbol '''dex''' represent the difference between powers of ten.

An order or factor of ten, dex is used both to refer to the function &lt;math&gt;dex(x) = 10^x&lt;/math&gt; and the number of (possibly fractional) orders of magnitude separating two numbers.  When dealing with [[wikt:log|log]] to the base 10 transform of a number set, the transform of 10, 100, and 1 000 000 is &lt;math&gt;\log_{10}(10) = 1&lt;/math&gt;, &lt;math&gt;\log_{10}(100) = 2&lt;/math&gt;, and &lt;math&gt;\log_{10}(1 000 000) = 6&lt;/math&gt;, so the difference between 10 and 100 in base 10 is 1 dex and the difference between 1 and 1 000 000 is 6 dex.

==Mathematics==
{{main|Mathematics}}
'''Def.''' an abstract representational system used in the study of [[wikt:number|number]]s, [[wikt:shape|shape]]s, [[wikt:structure|structure]] and [[wikt:change|change]] and the relationships between these concepts is called '''mathematics'''.

'''Def.''' the branches of mathematics used in the study of astronomy, [[astrophysics]] and cosmology is called '''astromathematics'''.

==Changes==
{{main|Changes}}
'''Def.''' the act or instance of making, or the process of becoming different is called '''change'''.

==Revolutions==
{{main|Changes/Revolutions|Revolutions}}
'''Def.''' the traversal of one body through an orbit around another body is called a '''revolution'''.

==Operations==
{{main|Operations}}
However, attempting to add 1 dome to 1 telescope may have little or no meaning. The operation of '''addition''' would be similar to the operation of '''construction'''.

If 1 G2V star is added to 1 M2V star the result may be a double star. The operation of addition here usually requires an explanation (a theory).
Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera that is attached to the telescope basically records a series of numbers - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.
==Inclinations==
{{main|Powers/Tendencies/Inclinations|Inclinations}}
[[Image:Orbit1.svg|thumb|right|200px|The diagram describes the parameters associated with orbital inclination (''i''). Credit: [[w:User:Lasunncty|Lasunncty]].]]
'''Def.''' the angle of intersection of a reference plane is called an '''inclination'''.

&quot;The orbital inclination [''i''] [of Mercury] varies between 5° and 10° with a 10&lt;sup&gt;6&lt;/sup&gt; yr period with smaller amplitude variations with a period of about 10&lt;sup&gt;5&lt;/sup&gt; yr.&quot;&lt;ref name=Peale/&gt;
{{clear}}

==Dimensional analyses==
{{main|Dimensional analyses}}
'''Def.''' a single aspect of a given thing is called a '''dimension'''.

Usually, in astronomy, a number is associated with a dimension or aspect of an entity. For example, the Earth is 1.50 x 10&lt;sup&gt;8&lt;/sup&gt; km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 10&lt;sup&gt;8&lt;/sup&gt; is a number.

'''Def.''' the study of the dimensions of quantities; used to obtain information about large complex systems, and as a means of checking equations is called '''dimensional analysis'''.

Prefixed values cannot be multiplied or divided together, and they have to be converted into non-prefixed standard form for such calculations. For example, 5&amp;nbsp;mV × 5&amp;nbsp;mA ≠ 25&amp;nbsp;mW. The correct calculation is: 5&amp;nbsp;mV × 5&amp;nbsp;mA = 5 × 10&lt;sup&gt;−3&lt;/sup&gt;&amp;nbsp;V × 5 × 10&lt;sup&gt;−3&lt;/sup&gt;&amp;nbsp;A = 25 x 10&lt;sup&gt;−6&lt;/sup&gt;&amp;nbsp;W = 25&amp;nbsp;µW = 0.025&amp;nbsp;mW.

Prefixes corresponding to an exponent that is divisible by three are often recommended. Hence &quot;100&amp;nbsp;m&quot; rather than &quot;1&amp;nbsp;hm&quot; (hectometre) or &quot;10&amp;nbsp;dam&quot; (decametres).  The &quot;non-three&quot; prefixes (hecto-, deca-, deci-, and centi-) are however more commonly used for everyday purposes than in science.

When units occur in exponentiation, for example, in square and cubic forms, any size prefix is considered part of the unit, and thus included in the exponentiation.
* {{gaps|1|km&lt;sup&gt;2&lt;/sup&gt;}} means one square kilometre or the size of a square of 1000&amp;nbsp;m by 1000&amp;nbsp;m and not 1000 [[w:square metre|square metre]]s.
* {{gaps|2|Mm&lt;sup&gt;3&lt;/sup&gt;}} means two cubic megametre or the size of two cubes of {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} or {{gaps|2|e=18|u=m&lt;sup&gt;3&lt;/sup&gt;}}, and not {{gaps|2|000|000|cubic metres}} ({{gaps|2|e=6|u=m&lt;sup&gt;3&lt;/sup&gt;}}).

;Examples
* {{gaps|5|u=cm}} = {{gaps|5|e=-2|u=m}} = {{gaps|5|&amp;times;|0.01|m}} = {{gaps|0.05|m}}
* {{gaps|3|u=MW}} = {{gaps|3|e=6|u=W}} = {{gaps|3|&amp;times;|1|000|000|W}} = {{gaps|3|000|000|W}}

==Astronomical units==
{{main|Astronomy/Units|Astronomical units}}
'''Def.''' &quot;1 day (d)&quot; is called the '''astronomical unit of time'''.&lt;ref name=Seidelmann&gt;{{ cite book
|author=P. K. Seidelmann
|title=Measuring the Universe The IAU and astronomical units
|publisher=International Astronomical Union
|date=1976
|url=http://www.iau.org/public/measuring/
|accessdate=2011-11-27 }}&lt;/ref&gt;

'''Def.''' &quot;the distance from the centre of the Sun at which a particle of negligible mass, in an unperturbed circular orbit, would have an orbital period of 365.2568983 days&quot; is called the '''Astronomical Unit''' (AU).&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;the distance at which one Astronomical Unit subtends an angle of one arcsecond&quot; is called the '''parsec''' (pc).&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;365.25 days&quot; is called a '''Julian year'''.&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;36,525 days&quot; is called a '''Julian century'''.&lt;ref name=Seidelmann/&gt;

{| class=&quot;wikitable&quot; style=&quot;float:center&quot;
|+ Units of Physics and Astronomy
|-
! Dimension !! Astronomy !! Symbol !! Physics !! Symbol !! Conversion 
|-
| time || 1 day || d || 1 second || s || 1 d = 86,400 s&lt;ref name=Seidelmann/&gt;
|-
| time || 1 &quot;Julian year&quot;&lt;ref name=Wilkins&gt;International Astronomical Union &quot;[http://www.iau.org/science/publications/proceedings_rules/units/ SI units]&quot; accessed February 18, 2010. (See Table 5 and section 5.15.) Reprinted from George A. Wilkins &amp; IAU Commission 5, [http://www.iau.org/static/publications/stylemanual1989.pdf &quot;The IAU Style Manual (1989)&quot;] (PDF file) in ''IAU Transactions'' Vol. XXB&lt;/ref&gt; || J || 1 second || s || 1 J = 31,557,600 s 
|-
| distance || 1 astronomical unit || AU || 1 meter || m || 1 AU = 149,597,870.691 km&lt;ref name=Seidelmann/&gt; 
|-
| mass || 1 Sun || M&lt;sub&gt;ʘ&lt;/sub&gt; || 1 kilogram || kg || 1 M&lt;sub&gt;ʘ&lt;/sub&gt; = 1.9891 x 10&lt;sup&gt;30&lt;/sup&gt; kg&lt;ref name=Seidelmann/&gt;
|-
| luminosity || 1 Sun || L&lt;sub&gt;ʘ&lt;/sub&gt; || 1 watt || W || 1 L&lt;sub&gt;ʘ&lt;/sub&gt; = 3.846 x 10&lt;sup&gt;26&lt;/sup&gt; W&lt;ref name=Williams2004&gt;{{ cite book
|author=David R. Williams
|title=Sun Fact Sheet
|publisher=NASA Goddard Space Flight Center
|location=Greenbelt, MD
|date=September 2004
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
|accessdate=2011-12-20 }}&lt;/ref&gt;
|-
| angular distance || 1 parsec || pc || 1 meter || m || 1 pc ~ 30.857 x 10&lt;sup&gt;12&lt;/sup&gt; km&lt;ref name=Seidelmann/&gt;
|-
|}

==Regions==
{{main|Spaces/Regions|Regions}}
[[Image:Center of the Milky Way Galaxy IV – Composite.jpg|thumb|right|250px|This is a composite image of the central region of our Milky Way galaxy. Credit: NASA/JPL-Caltech/ESA/CXC/STScI.]]
A '''region''' is a division or part of a space having definable characteristics but not always fixed boundaries.

'''Def.''' any [[wikt:considerable|considerable]] and [[wikt:connected|connected]] [[wikt:part|part]] of a [[wikt:space|space]] or [[wikt:surface|surface]]; specifically, a [[wikt:tract|tract]] of [[wikt:land|land]] or [[wikt:sea|sea]] of considerable but [[wikt:indefinite|indefinite]] [[wikt:extent|extent]]; a [[wikt:country|country]]; a [[wikt:district|district]]; in a broad sense, a place without special [[wikt:reference|reference]] to [[wikt:location|location]] or extent but viewed as an [[wikt:entity|entity]] for [[wikt:geographical|geographical]], [[wikt:social|social]] or [[wikt:cultural|cultural]] reasons is called a '''region'''.

'''Region''' is most commonly found as a term used in terrestrial and astrophysics sciences also an area, notably among the different sub-disciplines of [[geography]], studied by [[w:Regional geography|regional geographers]]. Regions consist of [[w:subregions|subregions]] that contain clusters of like areas that are distinctive by their uniformity of description based on a range of statistical data, for example demographic, and locales.

'''Def.''' a subset of &lt;math&gt;\R^n&lt;/math&gt; that is [[w:open set|open]] (in the standard [[w:Euclidean topology|Euclidean topology]]), [[w:connected set|connected]] and [[w:empty set|non-empty]] is called a '''region''', or '''region of &lt;math&gt;\R^n&lt;/math&gt;, where &lt;math&gt;\R^n&lt;/math&gt; is the n-dimensional real number system.'''
{{clear}}

==Areas==
{{main|Spaces/Areas|Areas}}
[[Image:RandLintegrals.png|thumb|right|250px|This diagram shows an approximation to an area under a curve. Credit: [[commons:User:Dubhe|Dubhe]].]]
In the figure on the right, an area is the difference in the x-direction times the difference in the y-direction.

This rectangle cornered at the origin of the curvature represents an area for the curve.
{{clear}}

==Orbits==
{{main|Spaces/Regions/Spheres/Orbits|Orbits}}
[[Image:Kepler orbits.svg|thumb|right|250px|A hyperbolic pass is indicated by the blue line with an eccentricity of 1.3. A parabolic pass is the green line. The elliptical orbit in red has an eccentricity (''e'') of 0.7. Credit: [[commons:User:Stamcose|Stamcose]].]]
[[Image:Orbit3.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with circular orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:Orbit5.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with elliptic orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:ISEE3-ICE-trajectory.gif|thumb|left|250px|ISEE-3 is inserted into a &quot;halo&quot; orbit on June 10, 1982. Credit: NASA.]]
'''Def.''' a circular or elliptical path of one object around another object is called an '''orbit'''.

Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of [[w:epicycles|epicycles]], which are the sums of numerous circular motions.&lt;ref&gt;''Encyclopaedia Britannica'', 1968, vol. 2, p. 645&lt;/ref&gt; Theories of this kind predicted paths of the planets moderately well, until [[w:Johannes Kepler|Johannes Kepler]] was able to show that the motions of planets were in fact (at least approximately) elliptical motions.&lt;ref name=Caspar&gt;M Caspar, ''Kepler'' (1959, Abelard-Schuman), at pp.131–140; A Koyré, ''The Astronomical Revolution: Copernicus, Kepler, Borelli'' (1973, Methuen), pp. 277–279&lt;/ref&gt;

In the [[w:geocentric model|geocentric model]] of the solar system, the [[w:celestial spheres|celestial spheres]] model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as [[w:deferent and epicycle|deferent and epicycle]]s were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.

In [[Astronomy/Theory|theoretical astronomy]], whether the Earth moves or not, serving as a fixed point with which to measure movements by objects or entities, or there is a [[w:solar system|solar system]] with the [[Stars/Sun|Sun]] near its center, is a matter of simplicity and calculational accuracy. Copernicus's theory provided a strikingly simple explanation for the apparent retrograde motions of the planets&amp;mdash;namely as [[w:parallax|parallactic]] displacements resulting from the Earth's motion around the Sun&amp;mdash;an important consideration in [[w:Johannes Kepler|Johannes Kepler]]'s conviction that the theory was substantially correct.&lt;ref name=Linton&gt;{{ cite book
|author=Christopher M. Linton
|title=From Eudoxus to Einstein&amp;mdash;A History of Mathematical Astronomy
|publisher=Cambridge University Press
|location=Cambridge
|date=2004
|editor=
|pages=
|url=
|bibcode=
|doi=
|pmid=
|isbn=978-0-521-82750-8
}}&lt;/ref&gt; &quot;[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy&quot;&lt;ref name=Linton/&gt; with the Earth at the center. Using a computer, this means that for competing programs, one written for each theory, the heliocentric program finishes first (for a mutually specified high degree of accuracy).

Orbits come in many shapes and motions. The simplest forms are a circle or an ellipse.
{{clear}}

==Infinitesimals==
{{main|Infinitesimals}}
'''Notation''': let the symbol &lt;math&gt;d&lt;/math&gt; represent an '''infinitesimal difference in'''.

'''Notation''': let the symbol &lt;math&gt;\partial&lt;/math&gt; represent an '''infinitesimal difference in''' one of more than one.

==Distances==
{{main|Distances}}
[[Image:Distancedisplacement.svg|thumb|right|250px|Distance along a path is compared in this diagram with displacement. Credit: .]]
'''Def.''' the amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line is called a '''distance'''.

'''Distance''' (or '''farness''') is a numerical description of how far apart objects are.  In [[physics]] or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. &quot;two counties over&quot;).  In [[mathematics]], a distance function or [[w:Metric (mathematics)|metric]] is a generalization of the concept of physical distance.  A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be &quot;close to&quot; or &quot;far away from&quot; each other.

'''Def.'''
# a series of interconnected rings or links usually made of metal,
# a series of interconnected links of known length, used as a measuring device,
# a long measuring tape,
# a unit of length equal to 22 yards. The length of a Gunter's surveying chain. The length of a cricket pitch. Equal to 20.12 metres. Equal to 4 rods. Equal to 100 links.
# a totally ordered set, especially a totally ordered subset of a poset,
# iron links bolted to the side of a vessel to bold the dead-eyes connected with the shrouds; also, the channels, or
# the warp threads of a web

is called a '''chain'''.

'''Def.''' a unit of length equal to 220 yards or exactly 201.168 meters, now only used in measuring distances in horse racing is called a '''furlong'''.

'''Def.'''
# a trench cut in the soil, as when plowed in order to plant a crop or
# any trench, channel, or groove, as in wood or metal

is called a '''furrow'''.

'''Def.''' the distance that a person can walk in one hour, commonly taken to be approximately three English miles (about five kilometers) is called a '''league'''.

:&lt;math&gt;R_{\odot} (equatorial) = 696,342 km&lt;/math&gt;

:&lt;math&gt;R_J (equatorial) = 71,492 km&lt;/math&gt;

:&lt;math&gt;R_S (equatorial) = 60,268 km&lt;/math&gt;

:&lt;math&gt;R_U (equatorial) = 25,559 km&lt;/math&gt;

:&lt;math&gt;R_N (equatorial) = 24,764 km&lt;/math&gt;

Then,

:&lt;math&gt;R_J (equatorial) = F_J * R_{\odot} (equatorial),&lt;/math&gt;
:&lt;math&gt;R_S (equatorial) = F_S * R_{\odot} (equatorial),&lt;/math&gt;
:&lt;math&gt;R_U (equatorial) = F_U * R_{\odot} (equatorial), and&lt;/math&gt;
:&lt;math&gt;R_N (equatorial) = F_N * R_{\odot} (equatorial).&lt;/math&gt;
{{clear}}

==Cosmic distance ladders==
{{main|Distances/Extragalactics|Cosmic distance ladders}}
The [[w:apparent magnitude|apparent magnitude]], or the magnitude as seen by the observer, can be used to determine the distance ''D'' to the object in kiloparsecs (where 1&amp;nbsp;kpc equals 1000&amp;nbsp;parsecs) as follows:

:&lt;math&gt;\begin{smallmatrix}5 \cdot \log_{10} \frac{D}{\mathrm{kpc}}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}&lt;/math&gt;

where ''m'' the apparent magnitude and ''M'' the absolute magnitude.

==Diameters==
{{main|Dimensions/Breadths/Diameters|Diameters}}
'''Def.''' the length of any straight line between two points on the circumference of a circle that passes through the centre/center of the circle is called a '''diameter'''.

==Arithmetic dimensional analysis==

Usually, pure [[arithmetic]] only involves numbers. But, when arithmetic is used in a science such as [[radiation astronomy]], dimensional analysis is also applicable.

To build an observatory usually requires adding components together.

For example: 1 dome + 1 telescope + 1 outbuilding + 1 control room + 1 laboratory + 1 observation room may = 1 observatory.

Yet,

: 1 + 1 + 1 + 1 + 1 + 1 = 6 components in 1 simple observatory.

==Obliquities==
{{main|Spaces/Obliquities|Obliquities}}
'''Def.''' the quality of being oblique in direction, deviating from the horizontal or vertical; or the angle created by such a deviation is called '''obliquity'''.

'''Axial tilt''' (also called '''obliquity''') is the angle between an object's [[w:Axis of rotation|rotational axis]], and a line [[w:Perpendicular|perpendicular]] to its [[w:Orbital plane (astronomy)|orbital plane]]. The planet [[w:Venus|Venus]] has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like [[Earth]]. The planet [[w:Uranus|Uranus]] is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the [[Stars/Sun|Sun]]. Hence the axial tilt of Uranus is 97°.&lt;ref name=Williams&gt;{{ cite book
|author=David R. Williams
|title=Planetary Fact Sheet Notes
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html }}&lt;/ref&gt;

The obliquity of the Earth's axis has a period of about 41,000 years.&lt;ref name=Hays&gt;{{ cite journal
|author=J. D. Hays
|author2=John Imbrie
|author3=N. J. Shackleton
|title=Variations in the Earth's Orbit: Pacemaker of the Ice Ages
|journal=Science
|month=December
|year=1976
|volume=194
|issue=4270
|pages=
|url=http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2011-11-08 }}&lt;/ref&gt;

==Inverses==
{{main|Inverses}}
'''Def.''' the set of points that map to a given point (or set of points) under a specified function is called an '''inverse image'''.

Under the function given by &lt;math&gt;f(x)=x^2&lt;/math&gt;, the '''inverse image''' of 4 is &lt;math&gt;\{-2,2\}&lt;/math&gt;, as is the '''inverse image''' of &lt;math&gt;\{4\}&lt;/math&gt;.

==Antapex==

'''Def.''' the point to which the Sun appears to be moving with respect to the local stars is called the '''solar apex'''.

An antapex is a point that an astronomical object's total motion is directed away from. It is opposite to the apex.

The '''local standard of rest''' or '''LSR'''  follows the mean motion of material in the [[w:Milky Way|Milky Way]] in the neighborhood of the [[Stars/Sun|Sun]].&lt;ref name= Shu&gt;{{cite book
|title=The Physical Universe
|author= Frank H Shu
|page= 261
|url=http://books.google.com/?id=v_6PbAfapSAC&amp;pg=PA261
|isbn=0935702059
|date=1982
|publisher=University Science Books }}&lt;/ref&gt; The path of this material is not precisely circular.&lt;ref name=Binney&gt;{{cite book
|title=Galactic Astronomy
|author=James Binney
|author2=Michael Merrifield
|url=http://books.google.com/?id=arYYRoYjKacC&amp;pg=PA536
|page= 536
|date=1998
|isbn=0691025657
|publisher=Princeton University Press }}&lt;/ref&gt; The Sun follows the '''solar circle''' ([[w:Ellipse#Eccentricity|eccentricity]] ''e'' &lt; 0.1 ) at a speed of about 220&amp;nbsp;km/s in a clockwise direction when viewed from the [[w:galactic coordinates|galactic north pole]] at a radius of ≈ 8 [[w:kiloparsec|kpc]] about the center of the galaxy near [[w:Sgr A*|Sgr A*]], and has only a slight  motion, towards the [[w:Solar apex|solar apex]], relative to the LSR.&lt;ref name=Reid&gt;{{ cite book
|url=http://books.google.com/?id=bP9hZqoIfhMC&amp;pg=PA19
|title=Mapping the Galaxy and Nearby Galaxies
|editor=F. Combes, Keiichi Wada
|date=2008
|publisher=Springer
|isbn=0387727671
|author= Mark Reid
|display-authors=etal
|chapter=Mapping the Milky Way and the Local Group
|pages=19–20 }}&lt;/ref&gt; The Sun's [[w:peculiar motion|peculiar motion]] relative to the LSR is 13.4 km/s.&lt;ref name=Binney1&gt;{{ cite book
|author=Binney J.
|author2=Merrifield M.
|title=op. cit.
|isbn=0691025657
|chapter=§10.6
}}&lt;/ref&gt;&lt;ref name=Mamajek&gt;{{cite journal
|title=On the distance to the Ophiuchus star-forming region
|journal=Astron. Nachr.
|volume=AN 329
|doi= 10.1002/asna.200710827
|arxiv=0709.0505
|year=2008
|page=12
|author=E.E. Mamajek
|bibcode = 2008AN....329...10M }}&lt;/ref&gt; The LSR velocity is anywhere from 202–241&amp;nbsp;km/s.&lt;ref name=Majewski&gt;{{ cite journal
|title=Precision Astrometry, Galactic Mergers, Halo Substructure and Local Dark Matter
|author=Steven R. Majewski
|journal=Proceedings of IAU Symposium 248
|arxiv=0801.4927
|year=2008
|bibcode = 2008IAUS..248..450M
|doi = 10.1017/S1743921308019790
|volume=3 }}&lt;/ref&gt;

==Algebras==
{{main|Algebras}}
'''Notation''': let the symbol '''*''' designate an as yet unspecified operation.

'''Notation''': let the symbol '''R''' designate an as yet unspecified relation.

'''Def.''' a system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols is called an '''algebra'''.

Fundamentally, [[Portal:Algebra|algebra]] uses letters to represent as yet unspecified [[numbers]]. The numbers may be [[Numbers/Integers|integers]], [[The Number System#Rational Numbers|rational numbers]], [[The Number System#Irrational Numbers|irrational numbers]], or any [[Real Numbers|real number]] or [[Complex Numbers|complex number]]. As an [[w:Experimentalist|experimentalist]], eventually you must find a way to change unspecified numbers into specified ones. But, as a theoretician, first you are free to leave the numbers in some algebraic form, then to have your theory tested by any experimentalist you need to relate the algebraic terms of your theory to real or complex numbers.

Consider the lower case letters of the English alphabet: a and n. The statement, &quot;a * n R an&quot;, contains the operation * (followed by) and the relation R (spells the word).

The manipulations of these symbols are performed using operations.

'''Def.''' a [[wikt:procedure|procedure]] for generating a [[wikt:value|value]] from one or more other values (the [[wikt:operand|operand]]s; the value for any particular [operand] is unique) is called an '''operation'''.

'''Notation''': let the symbol &lt;math&gt;\sum&lt;/math&gt; represent the summation of many terms.

'''Notation''': let the symbol &lt;math&gt;\Pi&lt;/math&gt; represent the product of many terms.

The results are recorded using statements of relation.

'''Def.''' a relation in which each element of the domain is associated with exactly one element of the codomain is called a '''function'''.

==Geometries==
{{main|Geometries}}
[[Image:Similar-geometric-shapes.svg|thumb|right|250px|Mathematics: Figures shown in the same color are similar. Credit: [[commons:User:Amada44|Amada44]].]]
'''Def.''' of geometrical figures including triangles, squares, ellipses, arcs and more complex figures, having the same shape but possibly different size, rotational orientation, and position; in particular, having corresponding angles equal and corresponding line segments proportional; such that one can be had from the other using a sequence of operations of rotation, translation and scaling is called '''similar'''.

'''Def.''' a branch of mathematics that studies solutions of systems of algebraic equations using both algebra and geometry is called '''algebraic geometry'''.

'''Def.''' a branch of mathematics that investigates properties of figures through the coordinates of their points is called '''analytic geometry'''.

'''Def.''' a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane is called '''projective geometry'''.

'''Def.''' a set along with a collection of finitary functions and relations is called a '''structure'''.

'''Def.'''
# a set of points with some added structure,
# distance between things,
# physical extent a range of values or locations across two or three dimensions,
# physical extent in all directions, seen as an attribute of the universe,
# a set of points, each of which is uniquely specified by a number (the dimensionality) of coordinates,
# a generalized construct or set whose members have some property in common; typically there will be a geometric metaphor allowing these members to be viewed as &quot;points&quot;,
# a gap; an empty place,
# a (chiefly empty) area or volume with set limits or boundaries,

is called a '''space'''.

The universe as often perceived may be described spatially, sometimes with plane [[geometry]], other occasions with [[Ideas in Geometry/Spherical Geometry|spherical geometry]].
{{clear}}

==Coordinates==
{{main|Measurements/Coordinates|Coordinates}}
[[Image:Cartesian-coordinate-system-with-circle.svg|thumb|right|250px|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (''x'' - ''a'')&lt;sup&gt;2&lt;/sup&gt; + (''y'' - ''b'')&lt;sup&gt;2&lt;/sup&gt; = ''r''&lt;sup&gt;2&lt;/sup&gt; where ''a'' and ''b'' are the coordinates of the center (''a'', ''b'') and ''r'' is the radius. Credit: [[w:User:345Kai|345Kai]].]]
A '''Cartesian coordinate system''' specifies each point uniquely in a plane by a pair of numerical '''coordinates''', which are the [positive and negative numbers] signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a ''coordinate axis'' or just ''axis'' of the system, and the point where they meet is its ''origin'', usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
{{clear}}

===Triclinic coordinates===
[[Image:Reseaux 3D aP.png|thumb|right|100px]] A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol &quot;⋀&quot; means &quot;angle between&quot;.
{{clear}}

===Monoclinic coordinates===
[[Image:Monoclinic cell.svg|thumb|right|100px]] In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°.
{{clear}}

===Orthorhombic coordinates===
[[Image:Reseaux 3D oP.png|thumb|right|100px]] In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c.
{{clear}}

===Tetragonal coordinates===
[[Image:Reseaux 3D tP-2011-03-12.png|thumb|right|100px]] A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c.
{{clear}}

===Rhombohedral coordinates===
[[Image:Rhombohedral.svg|thumb|right|100px]] A rhombohedral system has a = b = c and α = β = γ &lt; 120°, ≠ 90°.
{{clear}}

===Hexagonal coordinates===
[[Image:Reseaux 3D hP.png|thumb|right|100px]] A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°.
{{clear}}

==Triangles==
{{main|Spaces/Angularities/Triangles|Triangles}}
[[Image:Simple triangle.svg|thumb|right|200px|This diagram shows a regular '''triangle''', the geometric shape. Credit: [[commons:User:Dbc334|Dbc334]].]]
'''Def.''' a polygon with three sides and three angles is called a '''triangle'''.
{{clear}}

==Curvatures==
{{main|Spaces/Curvatures|Curvatures}}
The graph at the top of [[Astronomy/Mathematics#Areas|areas]] shows a curve or curvature.

==Conic sections==
{{main|Forms/Rotundity/Conics/Sections|Conic sections}}
[[Image:Conic Sections.svg|thumb|right|250px|alt=Diagram of conic sections|Conics are of three types: parabolas , ellipses, including circles, and hyperbolas. Credit: .]]
[[Image:Ellipse parameters 2.svg|left|thumb|300px|Conic parameters are shown in the case of an ellipse. Credit: .]]
'''Def.''' any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola is called a '''conic section'''.

In [[mathematics]], a '''conic section''' (or just '''conic''') is a [[w:curve|curve]] obtained as the intersection of a [[w:cone (geometry)|cone]] (more precisely, a right circular [[w:conical surface|conical surface]]) with a [[w:plane (mathematics)|plane]].

Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of ''a''&gt;''b'', for which the major axis is horizontal; for the reverse case, interchange the symbols ''a'' and ''b''. For the hyperbola the east-west opening case is given. In all cases, ''a'' and ''b'' are positive.)

{| class=&quot;wikitable&quot;
! conic section
! equation
! eccentricity (''e'')
! linear eccentricity (''c'')
! semi-latus rectum (''ℓ'')
! focal parameter (''p'')
|-
|  [[w:circle|circle]]
|| &lt;math&gt;x^2+y^2=a^2 \,&lt;/math&gt;
|| &lt;math&gt; 0 \,&lt;/math&gt;
|| &lt;math&gt; 0 \,&lt;/math&gt;
|| &lt;math&gt; a \,&lt;/math&gt;
|| &lt;math&gt; \infty&lt;/math&gt;
|-
|  [[w:ellipse|ellipse]]
|| &lt;math&gt;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1&lt;/math&gt;
|| &lt;math&gt;\sqrt{1-\frac{b^2}{a^2}}&lt;/math&gt;
|| &lt;math&gt;\sqrt{a^2-b^2}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{a}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{\sqrt{a^2-b^2}}&lt;/math&gt;
|-
|  [[w:parabola|parabola]]
|| &lt;math&gt;y^2=4ax \,&lt;/math&gt;
|| &lt;math&gt; 1 \,&lt;/math&gt;
|| &lt;math&gt; a \, &lt;/math&gt;
|| &lt;math&gt; 2a \,&lt;/math&gt;
|| &lt;math&gt; 2a \, &lt;/math&gt;
|-
|  [[Conic sections|hyperbola]]
|| &lt;math&gt;\frac{x^2}{a^2}-\frac{y^2}{b^2}=1&lt;/math&gt;
|| &lt;math&gt;\sqrt{1+\frac{b^2}{a^2}}&lt;/math&gt;
|| &lt;math&gt;\sqrt{a^2+b^2}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{a}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{\sqrt{a^2+b^2}}&lt;/math&gt;
|}

The general parabola equation with a vertical axis

:&lt;math&gt;ax^2+bx+c=0&lt;/math&gt;

is solved in terms of the constants ''a'', ''b'', and ''c'' for x by

:&lt;math&gt;x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.&lt;/math&gt;

The general conic equation in a Cartesian plane (x,y) is

:&lt;math&gt; A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,,&lt;/math&gt;

where

:&lt;math&gt; (AC - B^2/4)F + BED/4 - CD^2/4 - AE^2/4 \ne 0.&lt;/math&gt;

For parabolas,

:&lt;math&gt; B^2 = 4AC,&lt;/math&gt;

where ''A'' and ''C'' are not both zero.
{{clear}}

==Variations==
{{main|Spaces/Variations|Variations}}
'''Def.''' a partial change in the form, position, state, or qualities of a thing or a related but distinct thing is called a '''variation'''.

==Precessions==
{{main|Motions/Precessions|Precessions}}
'''Def.''' any of several slow changes in an astronomical body's rotational or orbital parameters such as the slow gyration of the [[Earth]]’s axis around the pole of the ecliptic is called a '''precession'''.

'''Def.''' the slow westward shift of the equinoxes along the plane of the ecliptic, resulting from precession of an object's axis of rotation, and causing the equinoxes to occur earlier each year is called the '''precession of the equinoxes'''.

The equinoxes of Earth precess with a period of about 21,000 years.&lt;ref name=Hays/&gt;

==Rotations==
{{main|Rotations}}
[[Image:Rotating Sphere.gif|right|thumb|250px|A [[w:polyhedron|polyhedron]] resembling a sphere rotating around an axis. Credit: [[w:User:BorisFromStockdale|BorisFromStockdale]].]]
'''Def.''' the act of turning around a centre or an axis is called a '''rotation'''.

A '''rotation''' is a [[w:circular motion|circular]] movement of an object around a ''center'' (or ''[[w:point (geometry)|point]]'') ''of rotation''. A [[w:Three-dimensional space|three-dimensional]] object rotates always around an imaginary [[w:Line (geometry)|line]] called a ''rotation axis''. If the axis is within the body, and passes through its [[w:center of mass|center of mass]] the body is said to rotate upon itself, or ''[[wikt:spin|spin]]''. A '''rotation''' about an external point, e.g. the [[Earth]] about the [[Stars/Sun|Sun]], is called a revolution or ''[[w:orbit|orbital revolution]]''.

Axes of rotation can be multiple:
# one-fold - ⨀, ⦺, ⧀
# two-fold - ⨸,
# three-fold - ▲,
# four-fold - ◈,
# five-fold - ✪, or
# six-fold - ✱.

Higher-fold axes of rotation are possible. As the number-fold of axes increases, the polyhedron approaches a circle. Or, in three dimensions, a sphere.
{{clear}}

==Mirror planes==
{{main|Mirror planes}}
A mirror plane reflects on the other side the handedness that is on the initial side:
# ⨴ | ⨵, the plane between is the mirror so that on either side is the reflection of the other, here an axis of rotation out of the plane of the paper could place the reflection on top of the object on the other side of the mirror plane,
# ∀ | ∀, here such an axis of rotation would not work,
# ⊆ | ⊇, this one is like number two,
# ⊕ | ⊕, here rotational symmetry is preserved, and
# ⨫ | ⨬, here rotation axes exist in the plane of the paper.

==Resonances==
{{main|Resonances}}
[[Image:Galilean moon Laplace resonance animation.gif|right|thumb|365px|The [[w:Laplace resonance|Laplace resonance]]s of Ganymede, [[Rocks/Ice sheets/Europa|Europa]], and [[Io]] is illustrated. Credit: User:Matma Rex.]]
An orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other body doing the &quot;pushing&quot; will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of [[Jupiter]]'s moons [[Rocks/Rocky objects/Ganymede|Ganymede]], [[Rocks/Ice sheets/Europa|Europa]] and [[Io]], and the 2:3 resonance between [[Pluto]] and [[Neptune]]. Unstable resonances with [[Saturn]]'s inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large [[Solar System]] bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.
{{clear}}

==Eccentricities==
{{main|Eccentricities}}
'''Def.''' the ratio, constant for any particular conic section, of the distance of a point from the focus to its distance from the [[wikt:directrix|directrix]] is called the '''eccentricity'''.

For an ellipse, the eccentricity is the ratio of the distance from the center to a focus divided by the length of the semi-major axis.

&quot;Mercury's orbit eccentricity [''e''] varies between about 0.11 and 0.24 with the shortest time lapse between the extremes being about 4 x 10&lt;sup&gt;5&lt;/sup&gt; yr&quot;.&lt;ref name=Peale&gt;{{ cite journal
|author=Peale, S. J.
|title=Possible histories of the obliquity of Mercury
|journal=Astronomical Journal
|month=June
|year=1974
|volume=79
|issue=6
|pages=722-44
|bibcode=1974AJ.....79..722P
|doi=10.1086/111604
|pmid= }}&lt;/ref&gt; &quot;Smaller amplitude variations occur with about a 10&lt;sup&gt;5&lt;/sup&gt; yr period.&quot;&lt;ref name=Peale/&gt;

==Spherical geometries==
{{main|Geometries/Spheres|Spherical geometries}}
[[Image:Triangles (spherical geometry).jpg|thumb|right|250px|On a sphere, the sum of the angles of a triangle is not equal to 180°. Credit: .]]
'''Def.''' the [[wikt:non-Euclidean geometry|non-Euclidean geometry]] on the surface of a [[wikt:sphere|sphere]] is called '''spherical geometry'''.

'''Spherical geometry''' is the [[geometry]] of the two-[[w:dimension|dimension]]al surface of a [[sphere]]. It is an example of a [[geometry]] which is not Euclidean. Two practical applications of the principles of spherical geometry are to [[w:navigation|navigation]] and astronomy.

A sphere [suggested by the image of the Earth at right] is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional [[w:manifold|manifold]].

The '''great-circle''' or '''[[w:Great Circle|orthodromic]] distance''' is the shortest [[w:distance|distance]] between any two [[w:Point (geometry)|point]]s on the surface of a [[w:sphere|sphere]] measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is different from ordinary [[w:Euclidean geometry|Euclidean geometry]], the equations for distance take on a different form. The distance between two points in [[w:Euclidean space|Euclidean space]] is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In [[w:non-Euclidean geometry|non-Euclidean geometry]], straight lines are replaced with [[w:geodesic|geodesic]]s. Geodesics on the sphere are the ''[[w:great circle|great circle]]s'' (circles on the sphere whose centers are coincident with the center of the sphere).

Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the [[w:Riemannian circle|Riemannian circle]].

==Logical laws==
{{main|Maxims/Axioms/Logical laws|Logical laws}}
[[Image:Kepler laws diagram.svg|thumb|300px|The diagram illustrates Kepler's three laws using two planetary orbits. Credit: [[commons:User:Hankwang|Hankwang]].]]
Kepler's laws of planetary motion:
# The orbit of every planet is an [[w:ellipse|ellipse]] with the Sun at one of the two [[w:Focus (geometry)|foci]].
# A [[w:line (geometry)|line]] joining a planet and the Sun sweeps out equal [[w:area|area]]s during equal intervals of time.&lt;ref name=&quot;Wolfram2nd&quot;&gt;Bryant, Jeff; Pavlyk, Oleksandr. &quot;[http://demonstrations.wolfram.com/KeplersSecondLaw/ Kepler's Second Law]&quot;, ''Wolfram Demonstrations Project''. Retrieved December 27, 2009.&lt;/ref&gt;
# The [[w:square (algebra)|square]] of the [[w:orbital period|orbital period]] of a planet is directly [[w:Proportionality (mathematics)|proportional]] to the [[w:cube (arithmetic)|cube]] of the [[w:semi-major axis|semi-major axis]] of its orbit.

The diagram at the right illustrates Kepler's three laws of planetary orbits: (1) The orbits are ellipses, with focal points ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt; and ''&amp;fnof;''&lt;sub&gt;2&lt;/sub&gt; for the first planet and ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt; and ''&amp;fnof;''&lt;sub&gt;3&lt;/sub&gt; for the second planet. The Sun is placed in focal point ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt;. (2) The two shaded sectors ''A''&lt;sub&gt;1&lt;/sub&gt; and ''A''&lt;sub&gt;2&lt;/sub&gt; have the same surface area and the time for planet 1 to cover segment ''A''&lt;sub&gt;1&lt;/sub&gt; is equal to the time to cover segment ''A''&lt;sub&gt;2&lt;/sub&gt;. (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;3/2&lt;/sup&gt;&amp;nbsp;:&amp;nbsp;''a''&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;3/2&lt;/sup&gt;.

The simplest description of the paths astronomical objects may take when passing each other includes a hyperbolic and parabolic pass. When capture occurs it usually produces an elliptical orbit.

==Horizontal coordinate system==
{{main|Coordinates/Horizontals|Horizontal coordinate systems}}
[[Image:Horizontal coordinate system 2.svg|thumb|right|250px|This diagram describes altitude and azimuth. Credit: Francisco Javier Blanco González.]]
The altitude of an entity in the sky is given by the angle of the arc from the local horizon to the entity.

The horizontal coordinate system is a [[w:celestial coordinate system|celestial coordinate system]] that uses the observer's local [[w:horizon|horizon]] as the [[w:Fundamental plane (spherical coordinates)|fundamental plane]]. This coordinate system divides the sky into the upper [[w:sphere|hemisphere]] where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in the way. The [[w:Great circle|great circle]] separating hemispheres [is] called [the] celestial horizon or rational horizon. The pole of the upper hemisphere is called the [[w:Zenith|zenith]]. The pole of the lower hemisphere is called the [[w:Nadir|nadir]].&lt;ref name=Schombert&gt;{{ cite book
|url=http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html
|title=Earth Coordinate System
|author=James Schombert
|publisher=University of Oregon Department of Physics
|accessdate=19 March 2011 }}&lt;/ref&gt;

The horizontal coordinates are:
* '''Altitude (Alt)''', sometimes referred to as [[w:elevation (disambiguation) | elevation]],  is the angle between the object and the observer's local horizon. It is expressed as an angle between 0 degrees to 90 degrees.
* '''[[w:Azimuth|Azimuth]] (Az)''', that is the angle of the object around the horizon, usually measured from the north increasing towards the east. 
* '''Zenith distance''', the distance from directly overhead (i.e. the zenith) is sometimes used instead of altitude in some calculations using these coordinates.  The zenith distance is the [[w:complementary angles|complement]] of altitude (i.e. 90°-altitude).

==Fixed point in the sky==
[[Image:EquatorialDecRA.png|thumb|100px|right|By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA). Credit: .]]
[[Image:AxialTiltObliquity.png|thumb|250px|right|Earth is shown as viewed from the Sun; the orbit direction is counter-clockwise (to the left). Description of the relations between axial tilt (or obliquity), rotation axis, plane of orbit, celestial equator and ecliptic. Credit: .]]
The observations require precise [[w:measurement|measurement]] and adaptations to the movements of the Earth, especially when and where, for a time, an object or entity is available.

With the creation of a geographical grid, an observer needs to be able to fix a point in the sky. From many observations within a period of stability, an observer notices that patterns of visual objects or entities in the night sky repeat. Further, a choice is available: is the Earth moving or are the star patterns moving? Depending on latitude, the observer may have noticed that the days vary in length and the pattern of variation repeats after some number of days and nights. By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA).

Once these can be determined, the apparent absolute positions of objects or entities are available in a communicable form. The repeat pattern of (day/night)s allows the observer to calculate the RA and Dec at any point during the cycle for a new object, or approximations are made using RA and Dec for recognized objects.

Independent of the choice made (Earth moves or not), the pattern of objects is the same for days or nights of the repeating length once a year. The '''[[w:Equinox|vernal equinox]]''' is a day/night of equal length and the same pattern of objects in the night sky. The '''autumnal equinox''' is the other equal length day/night with its own pattern of objects in the night sky.

The projection of the Earth's equator and poles of rotation, or if the observer hasn't concluded as yet that it's the Earth that's rotating, the circulating pattern of stars in ever smaller circles heading in specific directions, is the celestial sphere.
{{clear}}

==Trigonometries==
{{main|Trigonometries}}
'''Def.''' the relationships between the [[wikt:side|side]]s and the [[wikt:angle|angle]]s of [[wikt:triangle|triangle]]s and the [[wikt:calculation|calculation]]s based on them is called '''trigonometry'''.

'''Trigonometry''' ... studies [[w:triangle|triangle]]s and the relationships between their sides and the angles between these sides. Trigonometry defines the [[w:trigonometric functions|trigonometric functions]], which describe those relationships and have applicability to cyclical phenomena, such as waves.

==Angular displacement==

For the speeds in units of ''c'', ''β'' = ''v''/''c'', &quot;[i]n the usual interpretation of superluminal motion, the apparent velocity is given by
:&lt;math&gt;\beta_{app} = { \beta_{jet} \sin \phi \over 1 - \beta_{jet} \cos \phi },&lt;/math&gt;

where ''β''&lt;sub&gt;jet&lt;/sub&gt;''c'' is the jet velocity, and the jet makes an angle ''Φ'' to the line of sight.&quot;&lt;ref name=Gabuzda&gt;{{ cite journal
|author=D. C. Gabuzda
|author2=J. F. C. Wardle
|author3=D. H. Roberts
|title=Superluminal motion in the BL Lacertae object OJ 287
|journal=The Astrophysical Journal
|month=January 15,
|year=1989
|volume=336
|issue=1
|pages=L59-62
|url=
|arxiv=
|bibcode=1989ApJ...336L..59G
|doi=10.1086/185361
|pmid=
|accessdate=2012-03-21 }}&lt;/ref&gt;

==Radius of the Earth==

Because the [[Earth]] is not perfectly spherical, no single value serves as its natural [[wikt:radius|radius]]. ''Earth radius'' is used as a unit of distance, especially in astronomy and [[geology]]. Any radius a distance from a point on the surface to the center falls between the polar minimum of about 6,357&amp;nbsp;km and the equatorial maximum of about 6,378&amp;nbsp;km (≈3,950 – 3,963&amp;nbsp;mi).

Equations for great-circle distance can be used to roughly calculate the shortest distance between points on the surface of the Earth (''as the crow flies''), and so have applications in [[w:navigation|navigation]].

Let &lt;math&gt;\phi_s,\lambda_s;\ \phi_f,\lambda_f\;\!&lt;/math&gt; be the geographical [[w:latitude|latitude]] and [[w:longitude|longitude]] of two points (a base &quot;standpoint&quot; and the destination &quot;forepoint&quot;), respectively, and &lt;math&gt;\Delta\phi,\Delta\lambda\;\!&lt;/math&gt; their absolute differences; then &lt;math&gt;\Delta\widehat{\sigma}\;\!&lt;/math&gt;, the [[w:central angle|central angle]] between them, is given by the [[w:spherical law of cosines|spherical law of cosines]]:

:&lt;math&gt;{\color{white}\Big|}\Delta\widehat{\sigma}=\arccos\big(\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda\big).\;\!&lt;/math&gt;

The distance ''d'', i.e. the [[w:arc length|arc length]], for a sphere of radius ''r'' and &lt;math&gt;\Delta \widehat{\sigma}\!&lt;/math&gt; given in

:&lt;math&gt;d = r \, \Delta\widehat{\sigma}.\,\!&lt;/math&gt;

This arccosine formula above can have large [[w:rounding error|rounding error]]s if the distance is small (if the two points are a kilometer apart the cosine of the central angle comes out 0.99999999). An equation known as the [[w:haversine formula|haversine formula]] is [[w:Condition number|numerically better-conditioned]] for small distances:&lt;ref name=Sinnott&gt;R.W. Sinnott, &quot;Virtues of the Haversine&quot;, Sky and Telescope, vol. 68, no. 2, 1984, p. 159&lt;/ref&gt;

A formula that is accurate for all distances is the following special case (a sphere, which is an ellipsoid with equal major and minor axes) of the [[w:Vincenty's formulae|Vincenty formula]] (which more generally is a method to compute distances on ellipsoids):&lt;ref name=Vincenty&gt;{{ cite journal
  | last = Vincenty
  | first = Thaddeus 
  | authorlink = Thaddeus Vincenty
  | title = Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations
  | journal = Survey Review
  | volume = 23
  | issue = 176
  | pages = 88–93
  | publisher = Directorate of Overseas Surveys
  | location = Kingston Road, Tolworth, Surrey
  | date = 1975-04-01
  | url = http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
  | format = PDF
  | accessdate = 2008-07-21 }}&lt;/ref&gt;

:&lt;math&gt;{\color{white}\frac{\bigg|}{|}|}\Delta\widehat{\sigma}=\arctan\left(\frac{\sqrt{\left(\cos\phi_f\sin\Delta\lambda\right)^2+\left(\cos\phi_s\sin\phi_f-\sin\phi_s\cos\phi_f\cos\Delta\lambda\right)^2}}{\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda}\right).\;\!&lt;/math&gt;

When programming a computer, one should use the &lt;code&gt;[[w:atan2|atan2]]()&lt;/code&gt; function rather than the ordinary arctangent function (&lt;code&gt;atan()&lt;/code&gt;), in order to simplify handling of the case where the denominator is zero, and to compute &lt;math&gt;\Delta\widehat{\sigma}\;\!&lt;/math&gt; unambiguously in all quadrants. Also, make sure that all latitudes and longitudes are in radians (rather than degrees) if that is what your programming language's sin(), cos() and atan2() functions expect (1 radian = 180 / &amp;pi; degrees, 1 degree = &amp;pi; / 180 radians).

==Distance computation==
[[Image:Stellarparallax2.svg|thumb|125px|right|The diagram describes [[w:Stellar parallax|Stellar parallax]] motion.]]
Distance measurement by parallax is a special case of the principle of [[w:triangulation|triangulation]], which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 [[w:arcsecond|arcsecond]],&lt;ref name=ZG44&gt;{{harvnb|Zeilik|Gregory|1998 | loc=p. 44}}.&lt;/ref&gt; leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see [[w:Parallax#Derivation|derivation]] below), the distance to an object (measured in [[w:parsec|parsec]]s) is the [[w:Reciprocal (mathematics)|reciprocal]] of the parallax (measured in [[w:arcsecond|arcsecond]]s): &lt;math&gt;d (\mathrm{pc}) = 1 / p (\mathrm{arcsec}).&lt;/math&gt; For example, the distance to [[w:Proxima Centauri|Proxima Centauri]] is 1/0.7687=1.3009 parsecs (4.243 ly).&lt;ref name=&quot;apj118&quot;&gt;{{cite journal
 | author=Benedict
 | title=Interferometric Astrometry of Proxima Centauri and Barnard's Star Using HUBBLE SPACE TELESCOPE Fine Guidance Sensor 3: Detection Limits for Substellar Companions
 | journal=The Astronomical Journal
 | year=1999 | volume=118 | issue=2 | pages=1086–1100
 | bibcode=1999astro.ph..5318B
 | doi=10.1086/300975
 | ref=harv |arxiv = astro-ph/9905318
 | author-separator=,
 | author2=G. Fritz
 | display-authors=2
 | last3=Chappell
 | first3=D. W.
 | last4=Nelan
 | first4=E.
 | last5=Jefferys
 | first5=W. H.
 | last6=Van Altena
 | first6=W.
 | last7=Lee
 | first7=J.
 | last8=Cornell
 | first8=D.
 | last9=Shelus
 | first9=P. J. }}&lt;/ref&gt;
{{clear}}

==Distance to the Moon==

Any distance to the Moon is often initially calculated as a multiple of the Earth radius &lt;math&gt;R_\oplus&lt;/math&gt;.

==Parallaxes==

'''Parallax''' is a displacement or difference in the [[w:apparent position|apparent position]] of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.&quot;&lt;ref name=Shorter&gt;{{ cite book
| quote=Mutual inclination of two lines meeting in an angle
| title=Shorter Oxford English Dictionary
| date=1968 }}&lt;/ref&gt;&quot;''Astron.'' Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object.&quot;&lt;ref name=Oxford&gt;{{ cite book
| title=Oxford English Dictionary
| date=1989
| edition=Second
| quote=
| url=http://dictionary.oed.com/cgi/entry/50171114?single=1&amp;query_type=word&amp;queryword=parallax&amp;first=1&amp;max_to_show=10 }}&lt;/ref&gt;

Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to celestial objects including to the [[Moon]], the [[Sun]], and to [[w:star|star]]s beyond the [[Solar System]].

==Diurnal parallax==

''Diurnal parallax'' is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the [[w:terrestrial planet|terrestrial planet]]s or [[w:asteroid|asteroid]]s seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.&quot;&lt;ref name=Seidelmann2005&gt;{{ cite book
| author=P. Kenneth Seidelmann
| date=2005
| title=Explanatory Supplement to the Astronomical Almanac
| publisher=University Science Books
| pages=123–125
| isbn=1891389459 }}&lt;/ref&gt;&lt;ref name=Barbieri&gt;{{ cite book
| author=Cesare Barbieri
| date=2007
| title=Fundamentals of astronomy
| pages=132–135
| publisher=CRC Press
| isbn=0750308869 }}&lt;/ref&gt;

==Lunar parallax==
[[Image:Lunaparallax.png|thumb|250px|right|Diagram of daily lunar parallax. Credit: .]]
[[Image:Lunarparallax 22 3 1988.png|thumb|right|250px|Example of lunar parallax: Occultation of Pleiades by the Moon. Credit: .]]
''Lunar parallax'' (often short for ''lunar horizontal parallax'' or ''lunar equatorial horizontal parallax''), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body,  has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.&lt;ref name=aa1981 /&gt;

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified.  Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface.  Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:-  one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram);  and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an   Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth&lt;ref&gt;Astronomical Almanac, e.g. for 1981: see Glossary; for formulae see Explanatory Supplement to the Astronomical Almanac, 1992, p.400&lt;/ref&gt; -- equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth.  The Earth-Moon linear distance varies continuously as the Moon follows its [[w:orbit of the moon|perturbed and approximately elliptical orbit]] around the Earth.  The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.&lt;ref name=aa1981&gt;Astronomical Almanac e.g. for 1981, section D&lt;/ref&gt; The [[w:Astronomical Almanac|Astronomical Almanac]] and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of [[w:lunar theory|lunar theory]].

Parallax can also be used to determine the distance to the [[Moon]].

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000&amp;nbsp;km. This procedure was first used by [[w:Aristarchus of Samos|Aristarchus of Samos]]&lt;ref name=Gutzwiller&gt;{{ cite journal
| doi=10.1103/RevModPhys.70.589
| title=Moon-Earth-Sun: The oldest three-body problem
| year=1998
| author=Gutzwiller, Martin C.
| journal=Reviews of Modern Physics
| volume=70
| issue=2
| pages=589
| ref=harv
| bibcode=1998RvMP...70..589G }}&lt;/ref&gt; and [[w:Hipparchus|Hipparchus]], and later found its way into the work of [[w:Ptolemy|Ptolemy]]. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:
:&lt;math&gt;\mathrm{distance}_{\textrm{moon}} = \frac {\mathrm{distance}_{\mathrm{observerbase}}} {\tan (\mathrm{angle})}&lt;/math&gt;

==Calculuses==
{{main|Calculuses|Calculus}}
'''Calculus''' uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

==Derivatives==
{{main|Effects/Derivatives|Derivatives}}
'''Def.''' a result of an operation of deducing one function from another according to some fixed law is called a '''derivative'''.

Let
: &lt;math&gt;y = f(x)&lt;/math&gt;
be a function where values of &lt;math&gt;x&lt;/math&gt; may be any real number and values resulting in &lt;math&gt;y&lt;/math&gt; are also any real number.

: &lt;math&gt;\Delta x&lt;/math&gt; is a small finite difference in &lt;math&gt;x&lt;/math&gt; which when put into the function &lt;math&gt;f(x)&lt;/math&gt; produces a &lt;math&gt;\Delta y&lt;/math&gt;.

These small differences can be manipulated with the operations of arithmetic: addition (&lt;math&gt;+&lt;/math&gt;), subtraction (&lt;math&gt;-&lt;/math&gt;), multiplication (&lt;math&gt;*&lt;/math&gt;), and division (&lt;math&gt;/&lt;/math&gt;).

: &lt;math&gt;\Delta y = f(x + \Delta x) - f(x)&lt;/math&gt;

Dividing &lt;math&gt;\Delta y&lt;/math&gt; by &lt;math&gt;\Delta x&lt;/math&gt; and taking the limit as &lt;math&gt;\Delta x&lt;/math&gt; → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

: &lt;math&gt;f(x) = x^2&lt;/math&gt;

: &lt;math&gt;f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + \Delta x^2&lt;/math&gt;

: &lt;math&gt;\Delta y/\Delta x = (x^2 + 2x\Delta x + \Delta x^2 - x^2)/\Delta x&lt;/math&gt;

: &lt;math&gt;\Delta y/\Delta x = 2x + \Delta x&lt;/math&gt;

as &lt;math&gt;\Delta x&lt;/math&gt; and&lt;math&gt;\Delta y&lt;/math&gt; go towards zero,

: &lt;math&gt;dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.&lt;/math&gt;

This ratio is called the derivative.

==Partial derivatives==
{{main|Effects/Derivatives/Partials|Partial derivatives}}
Let

: &lt;math&gt;y = f(x,z)&lt;/math&gt;

then

: &lt;math&gt;\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z&lt;/math&gt;

: &lt;math&gt;\partial y/ \partial x = \partial f(x,z)&lt;/math&gt;

where z is held constant and

: &lt;math&gt;\partial y / \partial z = \partial f(x,z)&lt;/math&gt;

where x is held constant.

==Gradients==
{{main|Spaces/Gradients|Gradients}}
'''Notation''': let the symbol &lt;math&gt;\nabla&lt;/math&gt; be the gradient, i.e., derivatives for multivariable functions.

: &lt;math&gt;\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.&lt;/math&gt;

==Area under a curve==

Consider the curve in the graph in the section about [[Astronomy/Mathematics#Areas|areas]]. The x-direction is left and right, the y-direction is vertical.

For

: &lt;math&gt;\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x&lt;/math&gt;

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

: &lt;math&gt;\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.&lt;/math&gt;

Any particular individual rectangle for a sum of rectangular areas is

: &lt;math&gt;f(x_i + \Delta x_i) * \Delta x_i.&lt;/math&gt;

The approximate area under the curve is the sum &lt;math&gt;\sum&lt;/math&gt; of all the individual (i) areas from i = 0 to as many as the area needed (n):

: &lt;math&gt;\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.&lt;/math&gt;

==Integrals==

'''Def.''' a number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed is called an '''integral'''.

'''Notation''': let the symbol &lt;math&gt;\int &lt;/math&gt; represent the '''integral'''.

: &lt;math&gt;limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.&lt;/math&gt;

This can be within a finite interval [a,b]

: &lt;math&gt;\int_a^b f(x) \; dx&lt;/math&gt;

when i = 0 the integral is evaluated at &lt;math&gt;a&lt;/math&gt; and i = n the integral is evaluated at &lt;math&gt;b&lt;/math&gt;. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

==Theoretical calculus==

'''Def.''' a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a '''calculus'''.

'''Calculus''' focuses on [[w:limit (mathematics)|limits]], [[w:function (mathematics)|functions]], [[w:derivative|derivative]]s, [[w:integral|integral]]s, and [[w:Series (mathematics)|infinite series]].

Although ''calculus'' (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero).

==Line integrals==

'''Def.''' an integral the domain of whose integrand is a curve is called a '''line integral'''.

&quot;The pulsar dispersion measures [(DM)] provide directly the value of

:&lt;math&gt;DM =  \int_0^\infty n_e\, ds&lt;/math&gt;

along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure&quot;&lt;ref name=Reynolds&gt;{{ cite journal
|author=R. J. Reynolds
|title=Line Integrals of n&lt;sub&gt;e&lt;/sub&gt; and &lt;math&gt;n_e^2&lt;/math&gt; at High Galactic Latitude
|journal=The Astrophysical Journal
|month=May 1,
|year=1991
|volume=372
|issue=05
|pages=L17-20
|url=http://adsabs.harvard.edu/full/1991ApJ...372L..17R
|arxiv=
|bibcode=1991ApJ...372L..17R
|doi=10.1086/186013
|pmid=
|accessdate=2013-12-17 }}&lt;/ref&gt;

: &lt;math&gt;EM = \int_0^\infty n_e^2 ds.&lt;/math&gt;

==Vectors==
{{main|Vectors}}
[[Image:3D Vector.svg|100px|thumb|right]] For standard basis, or unit, vectors ('''i''', '''j''', '''k''') there may be vector components of '''a''' ('''a'''&lt;sub&gt;x&lt;/sub&gt;, '''a'''&lt;sub&gt;y&lt;/sub&gt;, '''a'''&lt;sub&gt;z&lt;/sub&gt;).

'''Def.''' a directed quantity, one with both magnitude and direction; the signed difference between two points is called a '''vector'''.

&quot;An observed time series consists of ''N'' data values x(t&lt;sub&gt;α&lt;/sub&gt;) taken at a set of ''N'' discrete times {t&lt;sub&gt;α&lt;/sub&gt;}. Hence it defines an ''N''-dimensional ''contravariant vector'' in ''sampling space'', by taking as the α&lt;sup&gt;th&lt;/sup&gt; component of the vector, the value of the data at time t&lt;sub&gt;α&lt;/sub&gt;, i.e.,
:&lt;math&gt;x^\alpha = [x({t_1}),x({t_2}),...,x({t_N})].&lt;/math&gt;
This representation is the ''canonical basis'' for sampling space.&quot;&lt;ref name=Foster/&gt;
{{clear}}

==Tensors==
{{main|Tensors}}
'''Def.''' a mathematical object consisting of a set of components with n indices each of which range from 1 to m where n is the rank and m is the dimension is called a '''tensor'''.

&quot;An impressive array of time series analysis methods are equivalent to treating the data as a vector in function space, then projecting the data vector onto a subspace of low dimension. A geometric approach isolates and exposes many of the important features of time series techniques, directly adapts to irregular time spacing, and easily accommodates variable statistical weights. Tensor notation provides an ideal formalism for these techniques. It is quite convenient for distinguishing a variety of different vector spaces, and is the most compact notation for all the sums which arise in the analysis.&quot;&lt;ref name=Foster&gt;{{ cite journal
|author=Grant Foster
|title=Time Series Analysis by Projection. II. Tensor Methods for Time Series Analysis
|journal=The Astronomical Journal
|month=January
|year=1996
|volume=111
|issue=1
|pages=555-65
|url=http://adsabs.harvard.edu/full/1996AJ....111..555F
|arxiv=
|bibcode=1996AJ....111..555F
|doi=
|pmid=
|accessdate=2013-12-16 }}&lt;/ref&gt;

&quot;[T]he generally invariant line element

: &lt;math&gt; d s^2 = g_{\mu \nu} dx^{\mu}dx^{\nu} &lt;/math&gt;

[contains] the spacetime metric tensor &lt;math&gt;g_{\mu \nu} (x^{\rho}), \mu, \nu, \rho = 0, 1, 2, 3,&lt;/math&gt; [which] plays a dual role: on the one hand it determines the spacetime geometry, on the other it represents the (ten components of the) gravitational potential, and is thus a dynamical variable.&quot;&lt;ref name=Bicak&gt;{{ cite journal
|author=Jiří Bičák
|title=Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics, In: ''Einstein’s Field Equations and Their Physical Implications''
|journal=Lecture Notes in Physics
|publisher=Springer Berlin Heidelberg
|location=Berlin
|month=
|year=2000
|editor=
|volume=540
|issue=
|pages=1-126
|url=http://arxiv.org/pdf/gr-qc/0004016
|arxiv=
|bibcode=
|doi=10.1007/3-540-46580-4_1
|pmid=
|isbn=978-3-540-67073-5
|accessdate=2013-07-04 }}&lt;/ref&gt;

==Electronic computers==
{{main|Electronic computers}}
'''Def.''' a programmable electronic device that performs mathematical calculations and logical operations, especially one that can process, store and retrieve large amounts of data very quickly; now especially, a small one for personal or home use employed for manipulating text or graphics, accessing the Internet, or playing games or media is called a '''computer'''.

A '''computer''' is a general purpose device that can be [[w:Computer program|programmed]] to carry out a finite set of arithmetic or logical operations. Since a sequence of operations can be readily changed, the computer can solve more than one kind of problem.

==Programmings==
{{main|Programmings}}
A '''computer program''' (also '''[[w:computer software|software]]''', or just a '''program''') is a sequence of [[w:instruction (computer science)|instructions]] written to perform a specified task with a computer.&lt;ref name=&quot;pis-ch4-p132&quot;&gt;{{ cite book
  | last = Stair
  | first = Ralph M.
  |display-authors=etal
  | title = Principles of Information Systems, Sixth Edition
  | publisher = Thomson Learning, Inc.
  | date = 2003
  | pages = 132
  | isbn = 0-619-06489-7 }}&lt;/ref&gt; A computer requires programs to function, typically [[w:execution (computing)|executing]] the program's instructions in a [[w:central processing unit|central processor]].&lt;ref name=&quot;osc-ch3-p58&quot;&gt;{{ cite book
  | last = Silberschatz
  | first = Abraham
  | title = Operating System Concepts, Fourth Edition
  | publisher = Addison-Wesley
  | date = 1994
  | pages = 58
  | isbn = 0-201-50480-4 }}&lt;/ref&gt;

'''Computer programming''' (often shortened to '''programming''' or '''coding''') is the process of [[w:Software design|designing]], writing, [[w:Software testing|testing]], [[w:debugging|debugging]], and maintaining the [[w:source code|source code]] of [[w:computer program|computer program]]s.

==Probabilities==
{{main|Probabilities}}
'''Def.''' a number, between 0 and 1, expressing the precise likelihood of an event happening is called a '''probability'''.

'''Probability''' is a measure of the expectation that an event will occur or a statement is true. Probabilities are given a value between 0 (will not occur) and 1 (will occur).&lt;ref name=Feller&gt;{{ cite book
|author=William Feller
|date=1968
|title=An Introduction to Probability Theory and its Applications
|volume=1
|ISBN=0-471-25708-7
}}&lt;/ref&gt;  The higher the probability of an event, the more certain we are that the event will occur.

==Statistics==
{{main|Statistics}}
'''Def.''' a  mathematical science concerned with data collection, presentation, analysis, and interpretation is called '''statistics'''.

'''[[Statistics]]''' is the study of the collection, organization, analysis, interpretation, and presentation of [[w:data|data]].&lt;ref name=Dodge&gt;Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}}&lt;/ref&gt;&lt;ref&gt;[http://www.thefreedictionary.com/dict.asp?Word=statistics The Free Online Dictionary]&lt;/ref&gt; It deals with all aspects of this, including the planning of data collection in terms of the design of [[w:statistical survey|survey]]s and [[w:experimental design|experiments]].&lt;ref name=Dodge/&gt;

&quot;Statistics of projections are derived under a number of different null hypotheses.&quot;&lt;ref name=Foster/&gt;

==Hypotheses==
{{main|Hypotheses}}
# Each mathematical approach requires a proof of concept.

==See also==
{{div col|colwidth=12em}}
* [[w:Cosmic distance ladder|Cosmic distance ladder]]
* [[Portal:Euclidean geometry|Euclidean geometry]]
* [[Topic:Mathematical physics|Mathematical physics]]
* [[Portal:Mathematics|Mathematics]]
* [[Radiation astronomy/Courses/Principles|Principles of Radiation Astronomy]]
* [[Probability]]
* [[Radiation astronomy/Mathematics|Radiation mathematics]]
* [[Statistics]]
* [[Astronomy/Theory|Theoretical astronomy]]
* [[Radiation astronomy/Theory|Theoretical radiation astronomy]]
* [[Topic:Trigonometry|Trigonometry]]
{{Div col end}}

==References==
{{reflist|2}}

==Further reading==
* {{ cite book
|author=William Marshall Smart
|author2=Robin Michael Green
|title=Textbook on Spherical Astronomy, Sixth Edition
|publisher=University of Cambridge
|location=Cambridge
|date=July 7, 1977
|editor=
|pages=431
|url=http://books.google.com/books?id=W0f2vc2EePUC&amp;lr=&amp;source=gbs_navlinks_s
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=0 521 21516 1
|accessdate=2012-05-18 }}
* {{ cite journal 
| title=Large-scale expanding superstructures in galaxies 
| author=Tenorio-Tagle G
|author2=Bodenheimer P
| journal=Annual Review of Astronomy and Astrophysics 
| date=1988 
|volume=26 
|issue= 
|pages=145–97 
|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T 
}}

==External links==
* [http://www.bing.com/search?q=&amp;go=&amp;qs=n&amp;sk=&amp;sc=8-15&amp;qb=1&amp;FORM=AXRE Bing Advanced search]
* [http://books.google.com/ Google Books]
* [http://scholar.google.com/advanced_scholar_search?hl=en&amp;lr= Google scholar Advanced Scholar Search]
* [http://www.iau.org/ International Astronomical Union]
* [http://www.jstor.org/ JSTOR]
* [http://www.lycos.com/ Lycos search]
* [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED]
* [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center.]
* [http://www.questia.com/ Questia - The Online Library of Books and Journals]
* [http://online.sagepub.com/ SAGE journals online]
* [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System]
* [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search]
* [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer]
* [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database]
* [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA]
* [http://www.springerlink.com/ SpringerLink]
* [http://www.tandfonline.com/ Taylor &amp; Francis Online]
* [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter]
* [http://onlinelibrary.wiley.com/advanced/search Wiley Online Library Advanced Search]
* [http://search.yahoo.com/web/advanced Yahoo Advanced Web Search]

&lt;!-- footer templates --&gt;
{{Mathematics resources}}God played fortnite and liked math so much he added it to real life{{tlx|Principles of radiation astronomy}}{{tlx|Radiation astronomy resources}}{{Sisterlinks|Mathematical astronomy}}

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&lt;!-- categories --&gt;
[[Category:Mathematics/Resources]]
[[Category:Radiation astronomy/Resources]]
[[Category:Resources last modified in February 2019]]</text>
      <sha1>4zl37veq41rl0zoo7ucnlrpumvgnkd2</sha1>
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      <id>2624945</id>
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        <id>2888076</id>
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      <comment>Reverted 1 edit by [[Special:Contributions/204.219.240.33|204.219.240.33]] ([[User talk:204.219.240.33|talk]]) (TwinkleGlobal)</comment>
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      <text bytes="85649" xml:space="preserve">[[Image:Superclusters atlasoftheuniverse.gif|thumb|250px|right|Our universe within 1 billion light-years (307 Mpc) of Earth is shown to contain the local [[w:supercluster|supercluster]]s, [[w:galaxy filament|galaxy filament]]s and voids. Credit: Richard Powell.]]
Although most of the mathematics needed to understand the information acquired through astronomical observation comes from physics, there are special needs from situations that intertwine mathematics with phenomena that may not yet have sufficient physics to explain the observations. These two uses of mathematics make '''mathematical astronomy''' a continuing challenge.

Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera, specifically its charge-coupled device (CCD) detector, that is attached to the telescope basically converts or counts photons or electrons and records a series of numbers (the counts) - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.

An initial use of mathematics in astronomy is counting [[Radiation astronomy/Entities|entities]], sources, or objects in the sky.

Objects may be counted during the daytime or night.

One use of mathematics is the calculation of distance to an object in the sky.

{{clear}}

==Notations==

'''Notation''': let the symbol &lt;math&gt;\oplus&lt;/math&gt; indicate the [[Earth]].

'''Notation''': let the symbol '''ʘ''' or &lt;math&gt;\odot&lt;/math&gt; indicate the [[Stars/Sun|Sun]].

'''Notation''': let the symbol &lt;math&gt; I_{\odot} &lt;/math&gt; indicate the total solar irradiance.

'''Notation''': let the symbol &lt;math&gt; L_V &lt;/math&gt; indicate the solar visible luminosity.

'''Notation''': let the symbol &lt;math&gt; L_{\odot} &lt;/math&gt; indicate the solar bolometric luminosity.

'''Notation''': let the symbol &lt;math&gt; L_{bol} &lt;/math&gt; indicate the solar bolometric luminosity.

'''Notation''': let the symbol &lt;math&gt;M_{bol}&lt;/math&gt; represent the '''bolometric magnitude''', the total energy output.

'''Notation''': let the symbol &lt;math&gt;M_V&lt;/math&gt; represent the '''visual magnitude'''.

'''Notation''': let the symbol &lt;math&gt; M_{\odot} &lt;/math&gt; indicate the solar mass.

'''Notation''': let the symbol &lt;math&gt; Q_{\odot} &lt;/math&gt; represent the net solar charge.

'''Notation''': let the symbol &lt;math&gt;R_\oplus&lt;/math&gt; indicate the Earth's radius.

'''Notation''': let the symbol &lt;math&gt;R_J&lt;/math&gt; indicate the radius of Jupiter.

'''Notation''': let the symbol &lt;math&gt;R_{\odot}&lt;/math&gt; indicate the solar radius.

:{| border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;none&quot;
|-
! colspan=&quot;3&quot; |'''Notational locations'''
|-
|'''Weight'''
|'''Oversymbol'''
|'''Exponent'''
|-
|'''Coefficient'''
|'''Variable'''
|'''Operation'''
|-
|'''Number'''
|'''Range'''
|'''Index'''
|-
|}

For each of the notational locations around the central '''Variable''', conventions are often set by consensus as to use. For example, '''Exponent''' is often used as an exponent to a number or variable: 2&lt;sup&gt;-2&lt;/sup&gt; or x&lt;sup&gt;2&lt;/sup&gt;.

In the '''Notation'''s at the top of this section, '''Index''' is replaced by symbols for the Sun (ʘ), [[Earth]] (&lt;math&gt;R_\oplus&lt;/math&gt;), or can be for Jupiter (J) such as &lt;math&gt;R_J&lt;/math&gt;.

A common '''Oversymbol''' is one for the average &lt;math&gt;\overline{Variable}&lt;/math&gt;.

'''Operation''' may be replaced by a function, for example.

All notational locations could look something like

:{| border=&quot;0&quot; cellpadding=&quot;3&quot; cellspacing=&quot;0&quot; align=&quot;none&quot;
|-
|bx
|&lt;math&gt;-&lt;/math&gt;
|x = n
|-
|a
|&lt;math&gt;\sum&lt;/math&gt;
|f(x)
|-
|n
|→
|∞
|-
|}
where the center line means &quot;a x Σ f(x)&quot; for all added up values of f(x) when x = n from say 0 to infinity with each term in the sum before summation multiplied by bn, then divided by n for an average whenever n is finite.

==Abstractions==
{{main|Abstractions}}
A '''nomy''' (Latin ''nomia'') is a &quot;system of [[Laws|law]]s governing or [the] sum of [[knowledge]] regarding a (specified) field.&quot;&lt;ref name=Gove&gt;{{ cite book
|author=
|title=Webster's Seventh New Collegiate Dictionary
|publisher=G. &amp; C. Merriam Company
|location=Springfield, Massachusetts
|date=1963
|editor=Philip B. Gove
|pages=1221
|bibcode=
|doi=
|pmid=
|isbn=
}}&lt;/ref&gt; ''Nomology'' is the &quot;[[What is science?|science]] of physical and logical laws.&quot;&lt;ref name=Gove/&gt;

'''Def.''' the quality of dealing with ideas rather than events is called '''abstraction'''.

'''Def.''' the act of the theoretical way of looking at things; something that exists only in idealized form is called '''abstraction'''.

==Relations==
{{main|Relations}}
'''Notation''': let the relation symbol '''≠''' indicate that two expressions are different.

For example, 2 x 3 ≠ 5 x 7.

'''Notation''': let the relation symbol '''~''' represent '''similar to'''.

For example, depending on the scale involved, 7 ~ 8 on a scale of 10, 7/10 = 0.7 and 8/10 = 0.8. relative to numbers between 0.5 and 1.0, 0.7 ~ 0.8, but 0.2 ≁ 0.7.

Similarity may be close such as 0.7 ≈ 0.8, but 0.5 ~ 0.8. Or similarity may include equality, 5 ± 3 ≃ 4 ± 2. When the degree of equality is greater than the degree of similarity, the symbol ≅ is used. The reverse is represented by ≊.

==Differences==
{{main|Abstractions/Differences|Differences}}
Here's a [[Definitions/Theory#Theoretical definition|theoretical definition]]:

'''Def.''' an abstract relation between identity and sameness is called a '''difference'''.

'''Notation''': let the symbol &lt;math&gt;\Delta&lt;/math&gt; represent '''difference in'''.

==Order==
{{main|Order}}
Ordering numbers may mean listing them in increasing value. For example, 2 is less than 3 so that in increasing order 2,3 is the list.

'''Notation''': let the symbol '''&gt;''' represent '''greater than'''.

For example, the integer five (5) is greater than the integer (2): 5 &gt; 2.

'''Notation''': let the symbol '''&lt;''' represent '''less than'''.

For, example, 2 &lt; 3.

==Numbers==
{{main|Numbers}}
:{|style=&quot;border:1px solid #ddd; text-align:center; margin:auto&quot; cellspacing=&quot;20&quot;
|&lt;math&gt;1, 2, 3,\ldots\!&lt;/math&gt; || &lt;math&gt;\ldots,-2, -1, 0, 1, 2\,\ldots\!&lt;/math&gt; || &lt;math&gt; -2, \frac{2}{3}, 1.21\,\!&lt;/math&gt; || &lt;math&gt;-e, \sqrt{2}, 3, \pi\,\!&lt;/math&gt; || &lt;math&gt;2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!&lt;/math&gt;
|-
|[[Natural number]]s|| [[Integer]]s || [[w:Rational number|Rational number]]s || [[Real Numbers|Real number]]s || [[Complex Numbers|Complex number]]s
|}
[[Image:Avogadro's number in e notation.jpg|thumb|upright|200px|A calculator display showing an approximation to the [[w:Avogadro constant|Avogadro constant]] in E notation. Credit: [[commons:User:PRHaney|PRHaney]].]]
'''Scientific notation''' (more commonly known as '''standard form''') is a way of writing numbers that are too big or too small to be conveniently written in decimal form.  Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers.

{| class=&quot;wikitable&quot; style=&quot;float:left&quot;
|-
!Standard decimal notation
!Normalized scientific notation
|-
| 2
|{{val|2|e=0}}
|-
| 300
|{{val|3|e=2}}
|-
| 4,321.768
|{{val|4.321768|e=3}}
|-
| -53,000
|{{val|-5.3|e=4}}
|-
| 6,720,000,000
|{{val|6.72|e=9}}
|-
| 0.2
|{{val|2|e=-1}}
|-
| 0.000&amp;nbsp;000&amp;nbsp;007&amp;nbsp;51
|{{val|7.51|e=-9}}
|}

A '''metric prefix''' or '''SI prefix''' is a [[w:unit prefix|unit prefix]] that precedes a basic unit of measure to indicate a [[w:decimal|decadic]] [[w:multiple (mathematics)|multiple]] or [[w:fraction (mathematics)|fraction]] of the unit. Each prefix has a unique symbol that is prepended to the unit symbol.

{{SI prefixes}}

A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes [[w:Significant figures#Identifying significant digits|indicated to be significant.]]

Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures—1, 2, 3, 0, and 4; the two zeroes serve only as placeholders and add no precision to the original number.

When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but all of the place holding zeroes are incorporated into the exponent. Following these rules, 1,230,400 becomes 1.2304 x 10&lt;sup&gt;6&lt;/sup&gt;.

It is customary in scientific measurements to record all the significant digits from the measurements, for example, 1,230,400, but the measurement may have introduced an error which when calculated indicates the last significant digit has a range of values where the most likely one is the &quot;4&quot;. The range may be 3-5 so that the last significant digit plus this error may be written as (4,1) meaning 4-1=3 and 4+1=5.

Another example of significant digits is the speed of all [[w:massless particle|massless particle]]s and associated [[w:field (physics)|field]]s—including [[w:electromagnetic radiation|electromagnetic radiation]] such as [[w:light|light]]—in vacuum ... [The most accurate value is] 299792.4562±0.0011 [km/s].&lt;ref name=&quot;NIST heterodyne&quot;&gt;{{cite journal
 |last1=Evenson 
 |first1=KM 
 |display-authors=etal
 |year=1972
 |title=Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser
 |journal=Physical Review Letters
 |volume=29
 |issue=19 |pages=1346–49
 |doi=10.1103/PhysRevLett.29.1346
|bibcode=1972PhRvL..29.1346E
}}&lt;/ref&gt; The magnitude of the speed is 299792.4562 and the actual measured variation is ±0.0011 so that the last two significant digits &quot;62&quot; are most likely within a variation from &quot;51&quot; to &quot;73&quot;.

Most [[w:calculator|calculator]]s and many [[w:computer program|computer program]]s present very large and very small results in E notation. The [[w:E|letter ''E'' or ''e'']] is often used to represent ''times ten raised to the power of'' (which would be written as &quot;x&amp;nbsp;10&lt;sup&gt;''b''&lt;/sup&gt;&quot;) [where ''b'' represents a number] and is followed by the value of the exponent.

'''Def.''' any real number that cannot be expressed as a ratio of two integers is called an '''irrational number'''.

'''Def.''' incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof is called '''uncountable'''.

An uncountable set of numbers such as the irrational numbers lies somewhere between a finite set of numbers, for example, the set of natural factors of 6: {1,2,3,6}, and an infinite set of numbers such as the [[w:Natural number|natural number]]s.

'''Def.''' the branch of pure mathematics concerned with the properties of integers is called '''number theory'''.

==Arithmetics==
{{main|Arithmetics}}
Arithmetic involves the manipulation of numbers.

===Theory of arithmetic===

'''Def.''' the [[wikt:mathematics|mathematics]] of [[wikt:number|number]]s ([[wikt:integer|integer]]s, [[wikt:rational number|rational number]]s, [[wikt:real number|real number]]s, or [[wikt:complex number|complex number]]s) under the operations of [[wikt:addition|addition]], [[wikt:subtraction|subtraction]], [[wikt:multiplication|multiplication]], and [[wikt:division|division]] is called an '''arithmetic'''.

'''Def.''' a symbol ( = ) used in mathematics to indicate that two values are the same is called '''equals''', or '''equal to'''.

Consider the integers: 1 and 2. The statement, &quot;1 + 2 = 3&quot;, contains the operation + (addition) and the relation = (equals).

===Exponentials===
{{main|Exponentials}}
The exponential can require a different operator of arithmetic.

The number '''&lt;math&gt;e&lt;/math&gt;''' is an important [[w:mathematical constant|mathematical constant]], approximately equal to 2.71828, that is the base of the [[w:natural logarithm|natural logarithm]].&lt;ref&gt;Oxford English Dictionary, 2nd ed.: [http://oxforddictionaries.com/definition/english/natural%2Blogarithm natural logarithm]&lt;/ref&gt;

:&lt;math&gt;e = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots&lt;/math&gt; 

: e&lt;sup&gt;2&lt;/sup&gt; + e&lt;sup&gt;3&lt;/sup&gt; ≠ e&lt;sup&gt;5&lt;/sup&gt;. Yet
: e&lt;sup&gt;2&lt;/sup&gt; x e&lt;sup&gt;3&lt;/sup&gt; = e&lt;sup&gt;(2 + 3)&lt;/sup&gt; = e&lt;sup&gt;5&lt;/sup&gt;.

===Logarithms===
{{main|Logarithms}}
'''Def.''' For a number [...], the [exponent or power] to which a given ''base'' number must be raised in order to obtain [the number] is called a '''logarithm'''.

Consider 10&lt;sup&gt;3&lt;/sup&gt;, the ''base'' number is 10. The exponent is 3. The number itself is 1000. Using logarithm notation
: &lt;math&gt;\log_{10} 1000 = 3.&lt;/math&gt;

For 2&lt;sup&gt;4&lt;/sup&gt;,
: &lt;math&gt; \log_{2} 16 = 4.&lt;/math&gt;

For e&lt;sup&gt;4&lt;/sup&gt;,
: &lt;math&gt; \ln_{e} e^4 = 4.&lt;/math&gt;

'''Notation''': let the symbol '''dex''' represent the difference between powers of ten.

An order or factor of ten, dex is used both to refer to the function &lt;math&gt;dex(x) = 10^x&lt;/math&gt; and the number of (possibly fractional) orders of magnitude separating two numbers.  When dealing with [[wikt:log|log]] to the base 10 transform of a number set, the transform of 10, 100, and 1 000 000 is &lt;math&gt;\log_{10}(10) = 1&lt;/math&gt;, &lt;math&gt;\log_{10}(100) = 2&lt;/math&gt;, and &lt;math&gt;\log_{10}(1 000 000) = 6&lt;/math&gt;, so the difference between 10 and 100 in base 10 is 1 dex and the difference between 1 and 1 000 000 is 6 dex.

==Mathematics==
{{main|Mathematics}}
'''Def.''' an abstract representational system used in the study of [[wikt:number|number]]s, [[wikt:shape|shape]]s, [[wikt:structure|structure]] and [[wikt:change|change]] and the relationships between these concepts is called '''mathematics'''.

'''Def.''' the branches of mathematics used in the study of astronomy, [[astrophysics]] and cosmology is called '''astromathematics'''.

==Changes==
{{main|Changes}}
'''Def.''' the act or instance of making, or the process of becoming different is called '''change'''.

==Revolutions==
{{main|Changes/Revolutions|Revolutions}}
'''Def.''' the traversal of one body through an orbit around another body is called a '''revolution'''.

==Operations==
{{main|Operations}}
However, attempting to add 1 dome to 1 telescope may have little or no meaning. The operation of '''addition''' would be similar to the operation of '''construction'''.

If 1 G2V star is added to 1 M2V star the result may be a double star. The operation of addition here usually requires an explanation (a theory).
Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera that is attached to the telescope basically records a series of numbers - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.
==Inclinations==
{{main|Powers/Tendencies/Inclinations|Inclinations}}
[[Image:Orbit1.svg|thumb|right|200px|The diagram describes the parameters associated with orbital inclination (''i''). Credit: [[w:User:Lasunncty|Lasunncty]].]]
'''Def.''' the angle of intersection of a reference plane is called an '''inclination'''.

&quot;The orbital inclination [''i''] [of Mercury] varies between 5° and 10° with a 10&lt;sup&gt;6&lt;/sup&gt; yr period with smaller amplitude variations with a period of about 10&lt;sup&gt;5&lt;/sup&gt; yr.&quot;&lt;ref name=Peale/&gt;
{{clear}}

==Dimensional analyses==
{{main|Dimensional analyses}}
'''Def.''' a single aspect of a given thing is called a '''dimension'''.

Usually, in astronomy, a number is associated with a dimension or aspect of an entity. For example, the Earth is 1.50 x 10&lt;sup&gt;8&lt;/sup&gt; km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 10&lt;sup&gt;8&lt;/sup&gt; is a number.

'''Def.''' the study of the dimensions of quantities; used to obtain information about large complex systems, and as a means of checking equations is called '''dimensional analysis'''.

Prefixed values cannot be multiplied or divided together, and they have to be converted into non-prefixed standard form for such calculations. For example, 5&amp;nbsp;mV × 5&amp;nbsp;mA ≠ 25&amp;nbsp;mW. The correct calculation is: 5&amp;nbsp;mV × 5&amp;nbsp;mA = 5 × 10&lt;sup&gt;−3&lt;/sup&gt;&amp;nbsp;V × 5 × 10&lt;sup&gt;−3&lt;/sup&gt;&amp;nbsp;A = 25 x 10&lt;sup&gt;−6&lt;/sup&gt;&amp;nbsp;W = 25&amp;nbsp;µW = 0.025&amp;nbsp;mW.

Prefixes corresponding to an exponent that is divisible by three are often recommended. Hence &quot;100&amp;nbsp;m&quot; rather than &quot;1&amp;nbsp;hm&quot; (hectometre) or &quot;10&amp;nbsp;dam&quot; (decametres).  The &quot;non-three&quot; prefixes (hecto-, deca-, deci-, and centi-) are however more commonly used for everyday purposes than in science.

When units occur in exponentiation, for example, in square and cubic forms, any size prefix is considered part of the unit, and thus included in the exponentiation.
* {{gaps|1|km&lt;sup&gt;2&lt;/sup&gt;}} means one square kilometre or the size of a square of 1000&amp;nbsp;m by 1000&amp;nbsp;m and not 1000 [[w:square metre|square metre]]s.
* {{gaps|2|Mm&lt;sup&gt;3&lt;/sup&gt;}} means two cubic megametre or the size of two cubes of {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} or {{gaps|2|e=18|u=m&lt;sup&gt;3&lt;/sup&gt;}}, and not {{gaps|2|000|000|cubic metres}} ({{gaps|2|e=6|u=m&lt;sup&gt;3&lt;/sup&gt;}}).

;Examples
* {{gaps|5|u=cm}} = {{gaps|5|e=-2|u=m}} = {{gaps|5|&amp;times;|0.01|m}} = {{gaps|0.05|m}}
* {{gaps|3|u=MW}} = {{gaps|3|e=6|u=W}} = {{gaps|3|&amp;times;|1|000|000|W}} = {{gaps|3|000|000|W}}

==Astronomical units==
{{main|Astronomy/Units|Astronomical units}}
'''Def.''' &quot;1 day (d)&quot; is called the '''astronomical unit of time'''.&lt;ref name=Seidelmann&gt;{{ cite book
|author=P. K. Seidelmann
|title=Measuring the Universe The IAU and astronomical units
|publisher=International Astronomical Union
|date=1976
|url=http://www.iau.org/public/measuring/
|accessdate=2011-11-27 }}&lt;/ref&gt;

'''Def.''' &quot;the distance from the centre of the Sun at which a particle of negligible mass, in an unperturbed circular orbit, would have an orbital period of 365.2568983 days&quot; is called the '''Astronomical Unit''' (AU).&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;the distance at which one Astronomical Unit subtends an angle of one arcsecond&quot; is called the '''parsec''' (pc).&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;365.25 days&quot; is called a '''Julian year'''.&lt;ref name=Seidelmann/&gt;

'''Def.''' &quot;36,525 days&quot; is called a '''Julian century'''.&lt;ref name=Seidelmann/&gt;

{| class=&quot;wikitable&quot; style=&quot;float:center&quot;
|+ Units of Physics and Astronomy
|-
! Dimension !! Astronomy !! Symbol !! Physics !! Symbol !! Conversion 
|-
| time || 1 day || d || 1 second || s || 1 d = 86,400 s&lt;ref name=Seidelmann/&gt;
|-
| time || 1 &quot;Julian year&quot;&lt;ref name=Wilkins&gt;International Astronomical Union &quot;[http://www.iau.org/science/publications/proceedings_rules/units/ SI units]&quot; accessed February 18, 2010. (See Table 5 and section 5.15.) Reprinted from George A. Wilkins &amp; IAU Commission 5, [http://www.iau.org/static/publications/stylemanual1989.pdf &quot;The IAU Style Manual (1989)&quot;] (PDF file) in ''IAU Transactions'' Vol. XXB&lt;/ref&gt; || J || 1 second || s || 1 J = 31,557,600 s 
|-
| distance || 1 astronomical unit || AU || 1 meter || m || 1 AU = 149,597,870.691 km&lt;ref name=Seidelmann/&gt; 
|-
| mass || 1 Sun || M&lt;sub&gt;ʘ&lt;/sub&gt; || 1 kilogram || kg || 1 M&lt;sub&gt;ʘ&lt;/sub&gt; = 1.9891 x 10&lt;sup&gt;30&lt;/sup&gt; kg&lt;ref name=Seidelmann/&gt;
|-
| luminosity || 1 Sun || L&lt;sub&gt;ʘ&lt;/sub&gt; || 1 watt || W || 1 L&lt;sub&gt;ʘ&lt;/sub&gt; = 3.846 x 10&lt;sup&gt;26&lt;/sup&gt; W&lt;ref name=Williams2004&gt;{{ cite book
|author=David R. Williams
|title=Sun Fact Sheet
|publisher=NASA Goddard Space Flight Center
|location=Greenbelt, MD
|date=September 2004
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
|accessdate=2011-12-20 }}&lt;/ref&gt;
|-
| angular distance || 1 parsec || pc || 1 meter || m || 1 pc ~ 30.857 x 10&lt;sup&gt;12&lt;/sup&gt; km&lt;ref name=Seidelmann/&gt;
|-
|}

==Regions==
{{main|Spaces/Regions|Regions}}
[[Image:Center of the Milky Way Galaxy IV – Composite.jpg|thumb|right|250px|This is a composite image of the central region of our Milky Way galaxy. Credit: NASA/JPL-Caltech/ESA/CXC/STScI.]]
A '''region''' is a division or part of a space having definable characteristics but not always fixed boundaries.

'''Def.''' any [[wikt:considerable|considerable]] and [[wikt:connected|connected]] [[wikt:part|part]] of a [[wikt:space|space]] or [[wikt:surface|surface]]; specifically, a [[wikt:tract|tract]] of [[wikt:land|land]] or [[wikt:sea|sea]] of considerable but [[wikt:indefinite|indefinite]] [[wikt:extent|extent]]; a [[wikt:country|country]]; a [[wikt:district|district]]; in a broad sense, a place without special [[wikt:reference|reference]] to [[wikt:location|location]] or extent but viewed as an [[wikt:entity|entity]] for [[wikt:geographical|geographical]], [[wikt:social|social]] or [[wikt:cultural|cultural]] reasons is called a '''region'''.

'''Region''' is most commonly found as a term used in terrestrial and astrophysics sciences also an area, notably among the different sub-disciplines of [[geography]], studied by [[w:Regional geography|regional geographers]]. Regions consist of [[w:subregions|subregions]] that contain clusters of like areas that are distinctive by their uniformity of description based on a range of statistical data, for example demographic, and locales.

'''Def.''' a subset of &lt;math&gt;\R^n&lt;/math&gt; that is [[w:open set|open]] (in the standard [[w:Euclidean topology|Euclidean topology]]), [[w:connected set|connected]] and [[w:empty set|non-empty]] is called a '''region''', or '''region of &lt;math&gt;\R^n&lt;/math&gt;, where &lt;math&gt;\R^n&lt;/math&gt; is the n-dimensional real number system.'''
{{clear}}

==Areas==
{{main|Spaces/Areas|Areas}}
[[Image:RandLintegrals.png|thumb|right|250px|This diagram shows an approximation to an area under a curve. Credit: [[commons:User:Dubhe|Dubhe]].]]
In the figure on the right, an area is the difference in the x-direction times the difference in the y-direction.

This rectangle cornered at the origin of the curvature represents an area for the curve.
{{clear}}

==Orbits==
{{main|Spaces/Regions/Spheres/Orbits|Orbits}}
[[Image:Kepler orbits.svg|thumb|right|250px|A hyperbolic pass is indicated by the blue line with an eccentricity of 1.3. A parabolic pass is the green line. The elliptical orbit in red has an eccentricity (''e'') of 0.7. Credit: [[commons:User:Stamcose|Stamcose]].]]
[[Image:Orbit3.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with circular orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:Orbit5.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with elliptic orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:ISEE3-ICE-trajectory.gif|thumb|left|250px|ISEE-3 is inserted into a &quot;halo&quot; orbit on June 10, 1982. Credit: NASA.]]
'''Def.''' a circular or elliptical path of one object around another object is called an '''orbit'''.

Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of [[w:epicycles|epicycles]], which are the sums of numerous circular motions.&lt;ref&gt;''Encyclopaedia Britannica'', 1968, vol. 2, p. 645&lt;/ref&gt; Theories of this kind predicted paths of the planets moderately well, until [[w:Johannes Kepler|Johannes Kepler]] was able to show that the motions of planets were in fact (at least approximately) elliptical motions.&lt;ref name=Caspar&gt;M Caspar, ''Kepler'' (1959, Abelard-Schuman), at pp.131–140; A Koyré, ''The Astronomical Revolution: Copernicus, Kepler, Borelli'' (1973, Methuen), pp. 277–279&lt;/ref&gt;

In the [[w:geocentric model|geocentric model]] of the solar system, the [[w:celestial spheres|celestial spheres]] model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as [[w:deferent and epicycle|deferent and epicycle]]s were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.

In [[Astronomy/Theory|theoretical astronomy]], whether the Earth moves or not, serving as a fixed point with which to measure movements by objects or entities, or there is a [[w:solar system|solar system]] with the [[Stars/Sun|Sun]] near its center, is a matter of simplicity and calculational accuracy. Copernicus's theory provided a strikingly simple explanation for the apparent retrograde motions of the planets&amp;mdash;namely as [[w:parallax|parallactic]] displacements resulting from the Earth's motion around the Sun&amp;mdash;an important consideration in [[w:Johannes Kepler|Johannes Kepler]]'s conviction that the theory was substantially correct.&lt;ref name=Linton&gt;{{ cite book
|author=Christopher M. Linton
|title=From Eudoxus to Einstein&amp;mdash;A History of Mathematical Astronomy
|publisher=Cambridge University Press
|location=Cambridge
|date=2004
|editor=
|pages=
|url=
|bibcode=
|doi=
|pmid=
|isbn=978-0-521-82750-8
}}&lt;/ref&gt; &quot;[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy&quot;&lt;ref name=Linton/&gt; with the Earth at the center. Using a computer, this means that for competing programs, one written for each theory, the heliocentric program finishes first (for a mutually specified high degree of accuracy).

Orbits come in many shapes and motions. The simplest forms are a circle or an ellipse.
{{clear}}

==Infinitesimals==
{{main|Infinitesimals}}
'''Notation''': let the symbol &lt;math&gt;d&lt;/math&gt; represent an '''infinitesimal difference in'''.

'''Notation''': let the symbol &lt;math&gt;\partial&lt;/math&gt; represent an '''infinitesimal difference in''' one of more than one.

==Distances==
{{main|Distances}}
[[Image:Distancedisplacement.svg|thumb|right|250px|Distance along a path is compared in this diagram with displacement. Credit: .]]
'''Def.''' the amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line is called a '''distance'''.

'''Distance''' (or '''farness''') is a numerical description of how far apart objects are.  In [[physics]] or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. &quot;two counties over&quot;).  In [[mathematics]], a distance function or [[w:Metric (mathematics)|metric]] is a generalization of the concept of physical distance.  A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be &quot;close to&quot; or &quot;far away from&quot; each other.

'''Def.'''
# a series of interconnected rings or links usually made of metal,
# a series of interconnected links of known length, used as a measuring device,
# a long measuring tape,
# a unit of length equal to 22 yards. The length of a Gunter's surveying chain. The length of a cricket pitch. Equal to 20.12 metres. Equal to 4 rods. Equal to 100 links.
# a totally ordered set, especially a totally ordered subset of a poset,
# iron links bolted to the side of a vessel to bold the dead-eyes connected with the shrouds; also, the channels, or
# the warp threads of a web

is called a '''chain'''.

'''Def.''' a unit of length equal to 220 yards or exactly 201.168 meters, now only used in measuring distances in horse racing is called a '''furlong'''.

'''Def.'''
# a trench cut in the soil, as when plowed in order to plant a crop or
# any trench, channel, or groove, as in wood or metal

is called a '''furrow'''.

'''Def.''' the distance that a person can walk in one hour, commonly taken to be approximately three English miles (about five kilometers) is called a '''league'''.

:&lt;math&gt;R_{\odot} (equatorial) = 696,342 km&lt;/math&gt;

:&lt;math&gt;R_J (equatorial) = 71,492 km&lt;/math&gt;

:&lt;math&gt;R_S (equatorial) = 60,268 km&lt;/math&gt;

:&lt;math&gt;R_U (equatorial) = 25,559 km&lt;/math&gt;

:&lt;math&gt;R_N (equatorial) = 24,764 km&lt;/math&gt;

Then,

:&lt;math&gt;R_J (equatorial) = F_J * R_{\odot} (equatorial),&lt;/math&gt;
:&lt;math&gt;R_S (equatorial) = F_S * R_{\odot} (equatorial),&lt;/math&gt;
:&lt;math&gt;R_U (equatorial) = F_U * R_{\odot} (equatorial), and&lt;/math&gt;
:&lt;math&gt;R_N (equatorial) = F_N * R_{\odot} (equatorial).&lt;/math&gt;
{{clear}}

==Cosmic distance ladders==
{{main|Distances/Extragalactics|Cosmic distance ladders}}
The [[w:apparent magnitude|apparent magnitude]], or the magnitude as seen by the observer, can be used to determine the distance ''D'' to the object in kiloparsecs (where 1&amp;nbsp;kpc equals 1000&amp;nbsp;parsecs) as follows:

:&lt;math&gt;\begin{smallmatrix}5 \cdot \log_{10} \frac{D}{\mathrm{kpc}}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}&lt;/math&gt;

where ''m'' the apparent magnitude and ''M'' the absolute magnitude.

==Diameters==
{{main|Dimensions/Breadths/Diameters|Diameters}}
'''Def.''' the length of any straight line between two points on the circumference of a circle that passes through the centre/center of the circle is called a '''diameter'''.

==Arithmetic dimensional analysis==

Usually, pure [[arithmetic]] only involves numbers. But, when arithmetic is used in a science such as [[radiation astronomy]], dimensional analysis is also applicable.

To build an observatory usually requires adding components together.

For example: 1 dome + 1 telescope + 1 outbuilding + 1 control room + 1 laboratory + 1 observation room may = 1 observatory.

Yet,

: 1 + 1 + 1 + 1 + 1 + 1 = 6 components in 1 simple observatory.

==Obliquities==
{{main|Spaces/Obliquities|Obliquities}}
'''Def.''' the quality of being oblique in direction, deviating from the horizontal or vertical; or the angle created by such a deviation is called '''obliquity'''.

'''Axial tilt''' (also called '''obliquity''') is the angle between an object's [[w:Axis of rotation|rotational axis]], and a line [[w:Perpendicular|perpendicular]] to its [[w:Orbital plane (astronomy)|orbital plane]]. The planet [[w:Venus|Venus]] has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like [[Earth]]. The planet [[w:Uranus|Uranus]] is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the [[Stars/Sun|Sun]]. Hence the axial tilt of Uranus is 97°.&lt;ref name=Williams&gt;{{ cite book
|author=David R. Williams
|title=Planetary Fact Sheet Notes
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html }}&lt;/ref&gt;

The obliquity of the Earth's axis has a period of about 41,000 years.&lt;ref name=Hays&gt;{{ cite journal
|author=J. D. Hays
|author2=John Imbrie
|author3=N. J. Shackleton
|title=Variations in the Earth's Orbit: Pacemaker of the Ice Ages
|journal=Science
|month=December
|year=1976
|volume=194
|issue=4270
|pages=
|url=http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2011-11-08 }}&lt;/ref&gt;

==Inverses==
{{main|Inverses}}
'''Def.''' the set of points that map to a given point (or set of points) under a specified function is called an '''inverse image'''.

Under the function given by &lt;math&gt;f(x)=x^2&lt;/math&gt;, the '''inverse image''' of 4 is &lt;math&gt;\{-2,2\}&lt;/math&gt;, as is the '''inverse image''' of &lt;math&gt;\{4\}&lt;/math&gt;.

==Antapex==

'''Def.''' the point to which the Sun appears to be moving with respect to the local stars is called the '''solar apex'''.

An antapex is a point that an astronomical object's total motion is directed away from. It is opposite to the apex.

The '''local standard of rest''' or '''LSR'''  follows the mean motion of material in the [[w:Milky Way|Milky Way]] in the neighborhood of the [[Stars/Sun|Sun]].&lt;ref name= Shu&gt;{{cite book
|title=The Physical Universe
|author= Frank H Shu
|page= 261
|url=http://books.google.com/?id=v_6PbAfapSAC&amp;pg=PA261
|isbn=0935702059
|date=1982
|publisher=University Science Books }}&lt;/ref&gt; The path of this material is not precisely circular.&lt;ref name=Binney&gt;{{cite book
|title=Galactic Astronomy
|author=James Binney
|author2=Michael Merrifield
|url=http://books.google.com/?id=arYYRoYjKacC&amp;pg=PA536
|page= 536
|date=1998
|isbn=0691025657
|publisher=Princeton University Press }}&lt;/ref&gt; The Sun follows the '''solar circle''' ([[w:Ellipse#Eccentricity|eccentricity]] ''e'' &lt; 0.1 ) at a speed of about 220&amp;nbsp;km/s in a clockwise direction when viewed from the [[w:galactic coordinates|galactic north pole]] at a radius of ≈ 8 [[w:kiloparsec|kpc]] about the center of the galaxy near [[w:Sgr A*|Sgr A*]], and has only a slight  motion, towards the [[w:Solar apex|solar apex]], relative to the LSR.&lt;ref name=Reid&gt;{{ cite book
|url=http://books.google.com/?id=bP9hZqoIfhMC&amp;pg=PA19
|title=Mapping the Galaxy and Nearby Galaxies
|editor=F. Combes, Keiichi Wada
|date=2008
|publisher=Springer
|isbn=0387727671
|author= Mark Reid
|display-authors=etal
|chapter=Mapping the Milky Way and the Local Group
|pages=19–20 }}&lt;/ref&gt; The Sun's [[w:peculiar motion|peculiar motion]] relative to the LSR is 13.4 km/s.&lt;ref name=Binney1&gt;{{ cite book
|author=Binney J.
|author2=Merrifield M.
|title=op. cit.
|isbn=0691025657
|chapter=§10.6
}}&lt;/ref&gt;&lt;ref name=Mamajek&gt;{{cite journal
|title=On the distance to the Ophiuchus star-forming region
|journal=Astron. Nachr.
|volume=AN 329
|doi= 10.1002/asna.200710827
|arxiv=0709.0505
|year=2008
|page=12
|author=E.E. Mamajek
|bibcode = 2008AN....329...10M }}&lt;/ref&gt; The LSR velocity is anywhere from 202–241&amp;nbsp;km/s.&lt;ref name=Majewski&gt;{{ cite journal
|title=Precision Astrometry, Galactic Mergers, Halo Substructure and Local Dark Matter
|author=Steven R. Majewski
|journal=Proceedings of IAU Symposium 248
|arxiv=0801.4927
|year=2008
|bibcode = 2008IAUS..248..450M
|doi = 10.1017/S1743921308019790
|volume=3 }}&lt;/ref&gt;

==Algebras==
{{main|Algebras}}
'''Notation''': let the symbol '''*''' designate an as yet unspecified operation.

'''Notation''': let the symbol '''R''' designate an as yet unspecified relation.

'''Def.''' a system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols is called an '''algebra'''.

Fundamentally, [[Portal:Algebra|algebra]] uses letters to represent as yet unspecified [[numbers]]. The numbers may be [[Numbers/Integers|integers]], [[The Number System#Rational Numbers|rational numbers]], [[The Number System#Irrational Numbers|irrational numbers]], or any [[Real Numbers|real number]] or [[Complex Numbers|complex number]]. As an [[w:Experimentalist|experimentalist]], eventually you must find a way to change unspecified numbers into specified ones. But, as a theoretician, first you are free to leave the numbers in some algebraic form, then to have your theory tested by any experimentalist you need to relate the algebraic terms of your theory to real or complex numbers.

Consider the lower case letters of the English alphabet: a and n. The statement, &quot;a * n R an&quot;, contains the operation * (followed by) and the relation R (spells the word).

The manipulations of these symbols are performed using operations.

'''Def.''' a [[wikt:procedure|procedure]] for generating a [[wikt:value|value]] from one or more other values (the [[wikt:operand|operand]]s; the value for any particular [operand] is unique) is called an '''operation'''.

'''Notation''': let the symbol &lt;math&gt;\sum&lt;/math&gt; represent the summation of many terms.

'''Notation''': let the symbol &lt;math&gt;\Pi&lt;/math&gt; represent the product of many terms.

The results are recorded using statements of relation.

'''Def.''' a relation in which each element of the domain is associated with exactly one element of the codomain is called a '''function'''.

==Geometries==
{{main|Geometries}}
[[Image:Similar-geometric-shapes.svg|thumb|right|250px|Mathematics: Figures shown in the same color are similar. Credit: [[commons:User:Amada44|Amada44]].]]
'''Def.''' of geometrical figures including triangles, squares, ellipses, arcs and more complex figures, having the same shape but possibly different size, rotational orientation, and position; in particular, having corresponding angles equal and corresponding line segments proportional; such that one can be had from the other using a sequence of operations of rotation, translation and scaling is called '''similar'''.

'''Def.''' a branch of mathematics that studies solutions of systems of algebraic equations using both algebra and geometry is called '''algebraic geometry'''.

'''Def.''' a branch of mathematics that investigates properties of figures through the coordinates of their points is called '''analytic geometry'''.

'''Def.''' a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane is called '''projective geometry'''.

'''Def.''' a set along with a collection of finitary functions and relations is called a '''structure'''.

'''Def.'''
# a set of points with some added structure,
# distance between things,
# physical extent a range of values or locations across two or three dimensions,
# physical extent in all directions, seen as an attribute of the universe,
# a set of points, each of which is uniquely specified by a number (the dimensionality) of coordinates,
# a generalized construct or set whose members have some property in common; typically there will be a geometric metaphor allowing these members to be viewed as &quot;points&quot;,
# a gap; an empty place,
# a (chiefly empty) area or volume with set limits or boundaries,

is called a '''space'''.

The universe as often perceived may be described spatially, sometimes with plane [[geometry]], other occasions with [[Ideas in Geometry/Spherical Geometry|spherical geometry]].
{{clear}}

==Coordinates==
{{main|Measurements/Coordinates|Coordinates}}
[[Image:Cartesian-coordinate-system-with-circle.svg|thumb|right|250px|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (''x'' - ''a'')&lt;sup&gt;2&lt;/sup&gt; + (''y'' - ''b'')&lt;sup&gt;2&lt;/sup&gt; = ''r''&lt;sup&gt;2&lt;/sup&gt; where ''a'' and ''b'' are the coordinates of the center (''a'', ''b'') and ''r'' is the radius. Credit: [[w:User:345Kai|345Kai]].]]
A '''Cartesian coordinate system''' specifies each point uniquely in a plane by a pair of numerical '''coordinates''', which are the [positive and negative numbers] signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a ''coordinate axis'' or just ''axis'' of the system, and the point where they meet is its ''origin'', usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
{{clear}}

===Triclinic coordinates===
[[Image:Reseaux 3D aP.png|thumb|right|100px]] A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol &quot;⋀&quot; means &quot;angle between&quot;.
{{clear}}

===Monoclinic coordinates===
[[Image:Monoclinic cell.svg|thumb|right|100px]] In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°.
{{clear}}

===Orthorhombic coordinates===
[[Image:Reseaux 3D oP.png|thumb|right|100px]] In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c.
{{clear}}

===Tetragonal coordinates===
[[Image:Reseaux 3D tP-2011-03-12.png|thumb|right|100px]] A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c.
{{clear}}

===Rhombohedral coordinates===
[[Image:Rhombohedral.svg|thumb|right|100px]] A rhombohedral system has a = b = c and α = β = γ &lt; 120°, ≠ 90°.
{{clear}}

===Hexagonal coordinates===
[[Image:Reseaux 3D hP.png|thumb|right|100px]] A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°.
{{clear}}

==Triangles==
{{main|Spaces/Angularities/Triangles|Triangles}}
[[Image:Simple triangle.svg|thumb|right|200px|This diagram shows a regular '''triangle''', the geometric shape. Credit: [[commons:User:Dbc334|Dbc334]].]]
'''Def.''' a polygon with three sides and three angles is called a '''triangle'''.
{{clear}}

==Curvatures==
{{main|Spaces/Curvatures|Curvatures}}
The graph at the top of [[Astronomy/Mathematics#Areas|areas]] shows a curve or curvature.

==Conic sections==
{{main|Forms/Rotundity/Conics/Sections|Conic sections}}
[[Image:Conic Sections.svg|thumb|right|250px|alt=Diagram of conic sections|Conics are of three types: parabolas , ellipses, including circles, and hyperbolas. Credit: .]]
[[Image:Ellipse parameters 2.svg|left|thumb|300px|Conic parameters are shown in the case of an ellipse. Credit: .]]
'''Def.''' any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola is called a '''conic section'''.

In [[mathematics]], a '''conic section''' (or just '''conic''') is a [[w:curve|curve]] obtained as the intersection of a [[w:cone (geometry)|cone]] (more precisely, a right circular [[w:conical surface|conical surface]]) with a [[w:plane (mathematics)|plane]].

Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of ''a''&gt;''b'', for which the major axis is horizontal; for the reverse case, interchange the symbols ''a'' and ''b''. For the hyperbola the east-west opening case is given. In all cases, ''a'' and ''b'' are positive.)

{| class=&quot;wikitable&quot;
! conic section
! equation
! eccentricity (''e'')
! linear eccentricity (''c'')
! semi-latus rectum (''ℓ'')
! focal parameter (''p'')
|-
|  [[w:circle|circle]]
|| &lt;math&gt;x^2+y^2=a^2 \,&lt;/math&gt;
|| &lt;math&gt; 0 \,&lt;/math&gt;
|| &lt;math&gt; 0 \,&lt;/math&gt;
|| &lt;math&gt; a \,&lt;/math&gt;
|| &lt;math&gt; \infty&lt;/math&gt;
|-
|  [[w:ellipse|ellipse]]
|| &lt;math&gt;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1&lt;/math&gt;
|| &lt;math&gt;\sqrt{1-\frac{b^2}{a^2}}&lt;/math&gt;
|| &lt;math&gt;\sqrt{a^2-b^2}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{a}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{\sqrt{a^2-b^2}}&lt;/math&gt;
|-
|  [[w:parabola|parabola]]
|| &lt;math&gt;y^2=4ax \,&lt;/math&gt;
|| &lt;math&gt; 1 \,&lt;/math&gt;
|| &lt;math&gt; a \, &lt;/math&gt;
|| &lt;math&gt; 2a \,&lt;/math&gt;
|| &lt;math&gt; 2a \, &lt;/math&gt;
|-
|  [[Conic sections|hyperbola]]
|| &lt;math&gt;\frac{x^2}{a^2}-\frac{y^2}{b^2}=1&lt;/math&gt;
|| &lt;math&gt;\sqrt{1+\frac{b^2}{a^2}}&lt;/math&gt;
|| &lt;math&gt;\sqrt{a^2+b^2}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{a}&lt;/math&gt;
|| &lt;math&gt;\frac{b^2}{\sqrt{a^2+b^2}}&lt;/math&gt;
|}

The general parabola equation with a vertical axis

:&lt;math&gt;ax^2+bx+c=0&lt;/math&gt;

is solved in terms of the constants ''a'', ''b'', and ''c'' for x by

:&lt;math&gt;x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.&lt;/math&gt;

The general conic equation in a Cartesian plane (x,y) is

:&lt;math&gt; A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,,&lt;/math&gt;

where

:&lt;math&gt; (AC - B^2/4)F + BED/4 - CD^2/4 - AE^2/4 \ne 0.&lt;/math&gt;

For parabolas,

:&lt;math&gt; B^2 = 4AC,&lt;/math&gt;

where ''A'' and ''C'' are not both zero.
{{clear}}

==Variations==
{{main|Spaces/Variations|Variations}}
'''Def.''' a partial change in the form, position, state, or qualities of a thing or a related but distinct thing is called a '''variation'''.

==Precessions==
{{main|Motions/Precessions|Precessions}}
'''Def.''' any of several slow changes in an astronomical body's rotational or orbital parameters such as the slow gyration of the [[Earth]]’s axis around the pole of the ecliptic is called a '''precession'''.

'''Def.''' the slow westward shift of the equinoxes along the plane of the ecliptic, resulting from precession of an object's axis of rotation, and causing the equinoxes to occur earlier each year is called the '''precession of the equinoxes'''.

The equinoxes of Earth precess with a period of about 21,000 years.&lt;ref name=Hays/&gt;

==Rotations==
{{main|Rotations}}
[[Image:Rotating Sphere.gif|right|thumb|250px|A [[w:polyhedron|polyhedron]] resembling a sphere rotating around an axis. Credit: [[w:User:BorisFromStockdale|BorisFromStockdale]].]]
'''Def.''' the act of turning around a centre or an axis is called a '''rotation'''.

A '''rotation''' is a [[w:circular motion|circular]] movement of an object around a ''center'' (or ''[[w:point (geometry)|point]]'') ''of rotation''. A [[w:Three-dimensional space|three-dimensional]] object rotates always around an imaginary [[w:Line (geometry)|line]] called a ''rotation axis''. If the axis is within the body, and passes through its [[w:center of mass|center of mass]] the body is said to rotate upon itself, or ''[[wikt:spin|spin]]''. A '''rotation''' about an external point, e.g. the [[Earth]] about the [[Stars/Sun|Sun]], is called a revolution or ''[[w:orbit|orbital revolution]]''.

Axes of rotation can be multiple:
# one-fold - ⨀, ⦺, ⧀
# two-fold - ⨸,
# three-fold - ▲,
# four-fold - ◈,
# five-fold - ✪, or
# six-fold - ✱.

Higher-fold axes of rotation are possible. As the number-fold of axes increases, the polyhedron approaches a circle. Or, in three dimensions, a sphere.
{{clear}}

==Mirror planes==
{{main|Mirror planes}}
A mirror plane reflects on the other side the handedness that is on the initial side:
# ⨴ | ⨵, the plane between is the mirror so that on either side is the reflection of the other, here an axis of rotation out of the plane of the paper could place the reflection on top of the object on the other side of the mirror plane,
# ∀ | ∀, here such an axis of rotation would not work,
# ⊆ | ⊇, this one is like number two,
# ⊕ | ⊕, here rotational symmetry is preserved, and
# ⨫ | ⨬, here rotation axes exist in the plane of the paper.

==Resonances==
{{main|Resonances}}
[[Image:Galilean moon Laplace resonance animation.gif|right|thumb|365px|The [[w:Laplace resonance|Laplace resonance]]s of Ganymede, [[Rocks/Ice sheets/Europa|Europa]], and [[Io]] is illustrated. Credit: User:Matma Rex.]]
An orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other body doing the &quot;pushing&quot; will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of [[Jupiter]]'s moons [[Rocks/Rocky objects/Ganymede|Ganymede]], [[Rocks/Ice sheets/Europa|Europa]] and [[Io]], and the 2:3 resonance between [[Pluto]] and [[Neptune]]. Unstable resonances with [[Saturn]]'s inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large [[Solar System]] bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.
{{clear}}

==Eccentricities==
{{main|Eccentricities}}
'''Def.''' the ratio, constant for any particular conic section, of the distance of a point from the focus to its distance from the [[wikt:directrix|directrix]] is called the '''eccentricity'''.

For an ellipse, the eccentricity is the ratio of the distance from the center to a focus divided by the length of the semi-major axis.

&quot;Mercury's orbit eccentricity [''e''] varies between about 0.11 and 0.24 with the shortest time lapse between the extremes being about 4 x 10&lt;sup&gt;5&lt;/sup&gt; yr&quot;.&lt;ref name=Peale&gt;{{ cite journal
|author=Peale, S. J.
|title=Possible histories of the obliquity of Mercury
|journal=Astronomical Journal
|month=June
|year=1974
|volume=79
|issue=6
|pages=722-44
|bibcode=1974AJ.....79..722P
|doi=10.1086/111604
|pmid= }}&lt;/ref&gt; &quot;Smaller amplitude variations occur with about a 10&lt;sup&gt;5&lt;/sup&gt; yr period.&quot;&lt;ref name=Peale/&gt;

==Spherical geometries==
{{main|Geometries/Spheres|Spherical geometries}}
[[Image:Triangles (spherical geometry).jpg|thumb|right|250px|On a sphere, the sum of the angles of a triangle is not equal to 180°. Credit: .]]
'''Def.''' the [[wikt:non-Euclidean geometry|non-Euclidean geometry]] on the surface of a [[wikt:sphere|sphere]] is called '''spherical geometry'''.

'''Spherical geometry''' is the [[geometry]] of the two-[[w:dimension|dimension]]al surface of a [[sphere]]. It is an example of a [[geometry]] which is not Euclidean. Two practical applications of the principles of spherical geometry are to [[w:navigation|navigation]] and astronomy.

A sphere [suggested by the image of the Earth at right] is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional [[w:manifold|manifold]].

The '''great-circle''' or '''[[w:Great Circle|orthodromic]] distance''' is the shortest [[w:distance|distance]] between any two [[w:Point (geometry)|point]]s on the surface of a [[w:sphere|sphere]] measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is different from ordinary [[w:Euclidean geometry|Euclidean geometry]], the equations for distance take on a different form. The distance between two points in [[w:Euclidean space|Euclidean space]] is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In [[w:non-Euclidean geometry|non-Euclidean geometry]], straight lines are replaced with [[w:geodesic|geodesic]]s. Geodesics on the sphere are the ''[[w:great circle|great circle]]s'' (circles on the sphere whose centers are coincident with the center of the sphere).

Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the [[w:Riemannian circle|Riemannian circle]].

==Logical laws==
{{main|Maxims/Axioms/Logical laws|Logical laws}}
[[Image:Kepler laws diagram.svg|thumb|300px|The diagram illustrates Kepler's three laws using two planetary orbits. Credit: [[commons:User:Hankwang|Hankwang]].]]
Kepler's laws of planetary motion:
# The orbit of every planet is an [[w:ellipse|ellipse]] with the Sun at one of the two [[w:Focus (geometry)|foci]].
# A [[w:line (geometry)|line]] joining a planet and the Sun sweeps out equal [[w:area|area]]s during equal intervals of time.&lt;ref name=&quot;Wolfram2nd&quot;&gt;Bryant, Jeff; Pavlyk, Oleksandr. &quot;[http://demonstrations.wolfram.com/KeplersSecondLaw/ Kepler's Second Law]&quot;, ''Wolfram Demonstrations Project''. Retrieved December 27, 2009.&lt;/ref&gt;
# The [[w:square (algebra)|square]] of the [[w:orbital period|orbital period]] of a planet is directly [[w:Proportionality (mathematics)|proportional]] to the [[w:cube (arithmetic)|cube]] of the [[w:semi-major axis|semi-major axis]] of its orbit.

The diagram at the right illustrates Kepler's three laws of planetary orbits: (1) The orbits are ellipses, with focal points ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt; and ''&amp;fnof;''&lt;sub&gt;2&lt;/sub&gt; for the first planet and ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt; and ''&amp;fnof;''&lt;sub&gt;3&lt;/sub&gt; for the second planet. The Sun is placed in focal point ''&amp;fnof;''&lt;sub&gt;1&lt;/sub&gt;. (2) The two shaded sectors ''A''&lt;sub&gt;1&lt;/sub&gt; and ''A''&lt;sub&gt;2&lt;/sub&gt; have the same surface area and the time for planet 1 to cover segment ''A''&lt;sub&gt;1&lt;/sub&gt; is equal to the time to cover segment ''A''&lt;sub&gt;2&lt;/sub&gt;. (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;3/2&lt;/sup&gt;&amp;nbsp;:&amp;nbsp;''a''&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;3/2&lt;/sup&gt;.

The simplest description of the paths astronomical objects may take when passing each other includes a hyperbolic and parabolic pass. When capture occurs it usually produces an elliptical orbit.

==Horizontal coordinate system==
{{main|Coordinates/Horizontals|Horizontal coordinate systems}}
[[Image:Horizontal coordinate system 2.svg|thumb|right|250px|This diagram describes altitude and azimuth. Credit: Francisco Javier Blanco González.]]
The altitude of an entity in the sky is given by the angle of the arc from the local horizon to the entity.

The horizontal coordinate system is a [[w:celestial coordinate system|celestial coordinate system]] that uses the observer's local [[w:horizon|horizon]] as the [[w:Fundamental plane (spherical coordinates)|fundamental plane]]. This coordinate system divides the sky into the upper [[w:sphere|hemisphere]] where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in the way. The [[w:Great circle|great circle]] separating hemispheres [is] called [the] celestial horizon or rational horizon. The pole of the upper hemisphere is called the [[w:Zenith|zenith]]. The pole of the lower hemisphere is called the [[w:Nadir|nadir]].&lt;ref name=Schombert&gt;{{ cite book
|url=http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html
|title=Earth Coordinate System
|author=James Schombert
|publisher=University of Oregon Department of Physics
|accessdate=19 March 2011 }}&lt;/ref&gt;

The horizontal coordinates are:
* '''Altitude (Alt)''', sometimes referred to as [[w:elevation (disambiguation) | elevation]],  is the angle between the object and the observer's local horizon. It is expressed as an angle between 0 degrees to 90 degrees.
* '''[[w:Azimuth|Azimuth]] (Az)''', that is the angle of the object around the horizon, usually measured from the north increasing towards the east. 
* '''Zenith distance''', the distance from directly overhead (i.e. the zenith) is sometimes used instead of altitude in some calculations using these coordinates.  The zenith distance is the [[w:complementary angles|complement]] of altitude (i.e. 90°-altitude).

==Fixed point in the sky==
[[Image:EquatorialDecRA.png|thumb|100px|right|By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA). Credit: .]]
[[Image:AxialTiltObliquity.png|thumb|250px|right|Earth is shown as viewed from the Sun; the orbit direction is counter-clockwise (to the left). Description of the relations between axial tilt (or obliquity), rotation axis, plane of orbit, celestial equator and ecliptic. Credit: .]]
The observations require precise [[w:measurement|measurement]] and adaptations to the movements of the Earth, especially when and where, for a time, an object or entity is available.

With the creation of a geographical grid, an observer needs to be able to fix a point in the sky. From many observations within a period of stability, an observer notices that patterns of visual objects or entities in the night sky repeat. Further, a choice is available: is the Earth moving or are the star patterns moving? Depending on latitude, the observer may have noticed that the days vary in length and the pattern of variation repeats after some number of days and nights. By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA).

Once these can be determined, the apparent absolute positions of objects or entities are available in a communicable form. The repeat pattern of (day/night)s allows the observer to calculate the RA and Dec at any point during the cycle for a new object, or approximations are made using RA and Dec for recognized objects.

Independent of the choice made (Earth moves or not), the pattern of objects is the same for days or nights of the repeating length once a year. The '''[[w:Equinox|vernal equinox]]''' is a day/night of equal length and the same pattern of objects in the night sky. The '''autumnal equinox''' is the other equal length day/night with its own pattern of objects in the night sky.

The projection of the Earth's equator and poles of rotation, or if the observer hasn't concluded as yet that it's the Earth that's rotating, the circulating pattern of stars in ever smaller circles heading in specific directions, is the celestial sphere.
{{clear}}

==Trigonometries==
{{main|Trigonometries}}
'''Def.''' the relationships between the [[wikt:side|side]]s and the [[wikt:angle|angle]]s of [[wikt:triangle|triangle]]s and the [[wikt:calculation|calculation]]s based on them is called '''trigonometry'''.

'''Trigonometry''' ... studies [[w:triangle|triangle]]s and the relationships between their sides and the angles between these sides. Trigonometry defines the [[w:trigonometric functions|trigonometric functions]], which describe those relationships and have applicability to cyclical phenomena, such as waves.

==Angular displacement==

For the speeds in units of ''c'', ''β'' = ''v''/''c'', &quot;[i]n the usual interpretation of superluminal motion, the apparent velocity is given by
:&lt;math&gt;\beta_{app} = { \beta_{jet} \sin \phi \over 1 - \beta_{jet} \cos \phi },&lt;/math&gt;

where ''β''&lt;sub&gt;jet&lt;/sub&gt;''c'' is the jet velocity, and the jet makes an angle ''Φ'' to the line of sight.&quot;&lt;ref name=Gabuzda&gt;{{ cite journal
|author=D. C. Gabuzda
|author2=J. F. C. Wardle
|author3=D. H. Roberts
|title=Superluminal motion in the BL Lacertae object OJ 287
|journal=The Astrophysical Journal
|month=January 15,
|year=1989
|volume=336
|issue=1
|pages=L59-62
|url=
|arxiv=
|bibcode=1989ApJ...336L..59G
|doi=10.1086/185361
|pmid=
|accessdate=2012-03-21 }}&lt;/ref&gt;

==Radius of the Earth==

Because the [[Earth]] is not perfectly spherical, no single value serves as its natural [[wikt:radius|radius]]. ''Earth radius'' is used as a unit of distance, especially in astronomy and [[geology]]. Any radius a distance from a point on the surface to the center falls between the polar minimum of about 6,357&amp;nbsp;km and the equatorial maximum of about 6,378&amp;nbsp;km (≈3,950 – 3,963&amp;nbsp;mi).

Equations for great-circle distance can be used to roughly calculate the shortest distance between points on the surface of the Earth (''as the crow flies''), and so have applications in [[w:navigation|navigation]].

Let &lt;math&gt;\phi_s,\lambda_s;\ \phi_f,\lambda_f\;\!&lt;/math&gt; be the geographical [[w:latitude|latitude]] and [[w:longitude|longitude]] of two points (a base &quot;standpoint&quot; and the destination &quot;forepoint&quot;), respectively, and &lt;math&gt;\Delta\phi,\Delta\lambda\;\!&lt;/math&gt; their absolute differences; then &lt;math&gt;\Delta\widehat{\sigma}\;\!&lt;/math&gt;, the [[w:central angle|central angle]] between them, is given by the [[w:spherical law of cosines|spherical law of cosines]]:

:&lt;math&gt;{\color{white}\Big|}\Delta\widehat{\sigma}=\arccos\big(\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda\big).\;\!&lt;/math&gt;

The distance ''d'', i.e. the [[w:arc length|arc length]], for a sphere of radius ''r'' and &lt;math&gt;\Delta \widehat{\sigma}\!&lt;/math&gt; given in

:&lt;math&gt;d = r \, \Delta\widehat{\sigma}.\,\!&lt;/math&gt;

This arccosine formula above can have large [[w:rounding error|rounding error]]s if the distance is small (if the two points are a kilometer apart the cosine of the central angle comes out 0.99999999). An equation known as the [[w:haversine formula|haversine formula]] is [[w:Condition number|numerically better-conditioned]] for small distances:&lt;ref name=Sinnott&gt;R.W. Sinnott, &quot;Virtues of the Haversine&quot;, Sky and Telescope, vol. 68, no. 2, 1984, p. 159&lt;/ref&gt;

A formula that is accurate for all distances is the following special case (a sphere, which is an ellipsoid with equal major and minor axes) of the [[w:Vincenty's formulae|Vincenty formula]] (which more generally is a method to compute distances on ellipsoids):&lt;ref name=Vincenty&gt;{{ cite journal
  | last = Vincenty
  | first = Thaddeus 
  | authorlink = Thaddeus Vincenty
  | title = Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations
  | journal = Survey Review
  | volume = 23
  | issue = 176
  | pages = 88–93
  | publisher = Directorate of Overseas Surveys
  | location = Kingston Road, Tolworth, Surrey
  | date = 1975-04-01
  | url = http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
  | format = PDF
  | accessdate = 2008-07-21 }}&lt;/ref&gt;

:&lt;math&gt;{\color{white}\frac{\bigg|}{|}|}\Delta\widehat{\sigma}=\arctan\left(\frac{\sqrt{\left(\cos\phi_f\sin\Delta\lambda\right)^2+\left(\cos\phi_s\sin\phi_f-\sin\phi_s\cos\phi_f\cos\Delta\lambda\right)^2}}{\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda}\right).\;\!&lt;/math&gt;

When programming a computer, one should use the &lt;code&gt;[[w:atan2|atan2]]()&lt;/code&gt; function rather than the ordinary arctangent function (&lt;code&gt;atan()&lt;/code&gt;), in order to simplify handling of the case where the denominator is zero, and to compute &lt;math&gt;\Delta\widehat{\sigma}\;\!&lt;/math&gt; unambiguously in all quadrants. Also, make sure that all latitudes and longitudes are in radians (rather than degrees) if that is what your programming language's sin(), cos() and atan2() functions expect (1 radian = 180 / &amp;pi; degrees, 1 degree = &amp;pi; / 180 radians).

==Distance computation==
[[Image:Stellarparallax2.svg|thumb|125px|right|The diagram describes [[w:Stellar parallax|Stellar parallax]] motion.]]
Distance measurement by parallax is a special case of the principle of [[w:triangulation|triangulation]], which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 [[w:arcsecond|arcsecond]],&lt;ref name=ZG44&gt;{{harvnb|Zeilik|Gregory|1998 | loc=p. 44}}.&lt;/ref&gt; leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see [[w:Parallax#Derivation|derivation]] below), the distance to an object (measured in [[w:parsec|parsec]]s) is the [[w:Reciprocal (mathematics)|reciprocal]] of the parallax (measured in [[w:arcsecond|arcsecond]]s): &lt;math&gt;d (\mathrm{pc}) = 1 / p (\mathrm{arcsec}).&lt;/math&gt; For example, the distance to [[w:Proxima Centauri|Proxima Centauri]] is 1/0.7687=1.3009 parsecs (4.243 ly).&lt;ref name=&quot;apj118&quot;&gt;{{cite journal
 | author=Benedict
 | title=Interferometric Astrometry of Proxima Centauri and Barnard's Star Using HUBBLE SPACE TELESCOPE Fine Guidance Sensor 3: Detection Limits for Substellar Companions
 | journal=The Astronomical Journal
 | year=1999 | volume=118 | issue=2 | pages=1086–1100
 | bibcode=1999astro.ph..5318B
 | doi=10.1086/300975
 | ref=harv |arxiv = astro-ph/9905318
 | author-separator=,
 | author2=G. Fritz
 | display-authors=2
 | last3=Chappell
 | first3=D. W.
 | last4=Nelan
 | first4=E.
 | last5=Jefferys
 | first5=W. H.
 | last6=Van Altena
 | first6=W.
 | last7=Lee
 | first7=J.
 | last8=Cornell
 | first8=D.
 | last9=Shelus
 | first9=P. J. }}&lt;/ref&gt;
{{clear}}

==Distance to the Moon==

Any distance to the Moon is often initially calculated as a multiple of the Earth radius &lt;math&gt;R_\oplus&lt;/math&gt;.

==Parallaxes==

'''Parallax''' is a displacement or difference in the [[w:apparent position|apparent position]] of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.&quot;&lt;ref name=Shorter&gt;{{ cite book
| quote=Mutual inclination of two lines meeting in an angle
| title=Shorter Oxford English Dictionary
| date=1968 }}&lt;/ref&gt;&quot;''Astron.'' Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object.&quot;&lt;ref name=Oxford&gt;{{ cite book
| title=Oxford English Dictionary
| date=1989
| edition=Second
| quote=
| url=http://dictionary.oed.com/cgi/entry/50171114?single=1&amp;query_type=word&amp;queryword=parallax&amp;first=1&amp;max_to_show=10 }}&lt;/ref&gt;

Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to celestial objects including to the [[Moon]], the [[Sun]], and to [[w:star|star]]s beyond the [[Solar System]].

==Diurnal parallax==

''Diurnal parallax'' is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the [[w:terrestrial planet|terrestrial planet]]s or [[w:asteroid|asteroid]]s seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.&quot;&lt;ref name=Seidelmann2005&gt;{{ cite book
| author=P. Kenneth Seidelmann
| date=2005
| title=Explanatory Supplement to the Astronomical Almanac
| publisher=University Science Books
| pages=123–125
| isbn=1891389459 }}&lt;/ref&gt;&lt;ref name=Barbieri&gt;{{ cite book
| author=Cesare Barbieri
| date=2007
| title=Fundamentals of astronomy
| pages=132–135
| publisher=CRC Press
| isbn=0750308869 }}&lt;/ref&gt;

==Lunar parallax==
[[Image:Lunaparallax.png|thumb|250px|right|Diagram of daily lunar parallax. Credit: .]]
[[Image:Lunarparallax 22 3 1988.png|thumb|right|250px|Example of lunar parallax: Occultation of Pleiades by the Moon. Credit: .]]
''Lunar parallax'' (often short for ''lunar horizontal parallax'' or ''lunar equatorial horizontal parallax''), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body,  has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.&lt;ref name=aa1981 /&gt;

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified.  Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface.  Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:-  one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram);  and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an   Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth&lt;ref&gt;Astronomical Almanac, e.g. for 1981: see Glossary; for formulae see Explanatory Supplement to the Astronomical Almanac, 1992, p.400&lt;/ref&gt; -- equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth.  The Earth-Moon linear distance varies continuously as the Moon follows its [[w:orbit of the moon|perturbed and approximately elliptical orbit]] around the Earth.  The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.&lt;ref name=aa1981&gt;Astronomical Almanac e.g. for 1981, section D&lt;/ref&gt; The [[w:Astronomical Almanac|Astronomical Almanac]] and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of [[w:lunar theory|lunar theory]].

Parallax can also be used to determine the distance to the [[Moon]].

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000&amp;nbsp;km. This procedure was first used by [[w:Aristarchus of Samos|Aristarchus of Samos]]&lt;ref name=Gutzwiller&gt;{{ cite journal
| doi=10.1103/RevModPhys.70.589
| title=Moon-Earth-Sun: The oldest three-body problem
| year=1998
| author=Gutzwiller, Martin C.
| journal=Reviews of Modern Physics
| volume=70
| issue=2
| pages=589
| ref=harv
| bibcode=1998RvMP...70..589G }}&lt;/ref&gt; and [[w:Hipparchus|Hipparchus]], and later found its way into the work of [[w:Ptolemy|Ptolemy]]. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:
:&lt;math&gt;\mathrm{distance}_{\textrm{moon}} = \frac {\mathrm{distance}_{\mathrm{observerbase}}} {\tan (\mathrm{angle})}&lt;/math&gt;

==Calculuses==
{{main|Calculuses|Calculus}}
'''Calculus''' uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

==Derivatives==
{{main|Effects/Derivatives|Derivatives}}
'''Def.''' a result of an operation of deducing one function from another according to some fixed law is called a '''derivative'''.

Let
: &lt;math&gt;y = f(x)&lt;/math&gt;
be a function where values of &lt;math&gt;x&lt;/math&gt; may be any real number and values resulting in &lt;math&gt;y&lt;/math&gt; are also any real number.

: &lt;math&gt;\Delta x&lt;/math&gt; is a small finite difference in &lt;math&gt;x&lt;/math&gt; which when put into the function &lt;math&gt;f(x)&lt;/math&gt; produces a &lt;math&gt;\Delta y&lt;/math&gt;.

These small differences can be manipulated with the operations of arithmetic: addition (&lt;math&gt;+&lt;/math&gt;), subtraction (&lt;math&gt;-&lt;/math&gt;), multiplication (&lt;math&gt;*&lt;/math&gt;), and division (&lt;math&gt;/&lt;/math&gt;).

: &lt;math&gt;\Delta y = f(x + \Delta x) - f(x)&lt;/math&gt;

Dividing &lt;math&gt;\Delta y&lt;/math&gt; by &lt;math&gt;\Delta x&lt;/math&gt; and taking the limit as &lt;math&gt;\Delta x&lt;/math&gt; → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

: &lt;math&gt;f(x) = x^2&lt;/math&gt;

: &lt;math&gt;f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + \Delta x^2&lt;/math&gt;

: &lt;math&gt;\Delta y/\Delta x = (x^2 + 2x\Delta x + \Delta x^2 - x^2)/\Delta x&lt;/math&gt;

: &lt;math&gt;\Delta y/\Delta x = 2x + \Delta x&lt;/math&gt;

as &lt;math&gt;\Delta x&lt;/math&gt; and&lt;math&gt;\Delta y&lt;/math&gt; go towards zero,

: &lt;math&gt;dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.&lt;/math&gt;

This ratio is called the derivative.

==Partial derivatives==
{{main|Effects/Derivatives/Partials|Partial derivatives}}
Let

: &lt;math&gt;y = f(x,z)&lt;/math&gt;

then

: &lt;math&gt;\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z&lt;/math&gt;

: &lt;math&gt;\partial y/ \partial x = \partial f(x,z)&lt;/math&gt;

where z is held constant and

: &lt;math&gt;\partial y / \partial z = \partial f(x,z)&lt;/math&gt;

where x is held constant.

==Gradients==
{{main|Spaces/Gradients|Gradients}}
'''Notation''': let the symbol &lt;math&gt;\nabla&lt;/math&gt; be the gradient, i.e., derivatives for multivariable functions.

: &lt;math&gt;\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.&lt;/math&gt;

==Area under a curve==

Consider the curve in the graph in the section about [[Astronomy/Mathematics#Areas|areas]]. The x-direction is left and right, the y-direction is vertical.

For

: &lt;math&gt;\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x&lt;/math&gt;

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

: &lt;math&gt;\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.&lt;/math&gt;

Any particular individual rectangle for a sum of rectangular areas is

: &lt;math&gt;f(x_i + \Delta x_i) * \Delta x_i.&lt;/math&gt;

The approximate area under the curve is the sum &lt;math&gt;\sum&lt;/math&gt; of all the individual (i) areas from i = 0 to as many as the area needed (n):

: &lt;math&gt;\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.&lt;/math&gt;

==Integrals==

'''Def.''' a number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed is called an '''integral'''.

'''Notation''': let the symbol &lt;math&gt;\int &lt;/math&gt; represent the '''integral'''.

: &lt;math&gt;limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.&lt;/math&gt;

This can be within a finite interval [a,b]

: &lt;math&gt;\int_a^b f(x) \; dx&lt;/math&gt;

when i = 0 the integral is evaluated at &lt;math&gt;a&lt;/math&gt; and i = n the integral is evaluated at &lt;math&gt;b&lt;/math&gt;. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

==Theoretical calculus==

'''Def.''' a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a '''calculus'''.

'''Calculus''' focuses on [[w:limit (mathematics)|limits]], [[w:function (mathematics)|functions]], [[w:derivative|derivative]]s, [[w:integral|integral]]s, and [[w:Series (mathematics)|infinite series]].

Although ''calculus'' (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero).

==Line integrals==

'''Def.''' an integral the domain of whose integrand is a curve is called a '''line integral'''.

&quot;The pulsar dispersion measures [(DM)] provide directly the value of

:&lt;math&gt;DM =  \int_0^\infty n_e\, ds&lt;/math&gt;

along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure&quot;&lt;ref name=Reynolds&gt;{{ cite journal
|author=R. J. Reynolds
|title=Line Integrals of n&lt;sub&gt;e&lt;/sub&gt; and &lt;math&gt;n_e^2&lt;/math&gt; at High Galactic Latitude
|journal=The Astrophysical Journal
|month=May 1,
|year=1991
|volume=372
|issue=05
|pages=L17-20
|url=http://adsabs.harvard.edu/full/1991ApJ...372L..17R
|arxiv=
|bibcode=1991ApJ...372L..17R
|doi=10.1086/186013
|pmid=
|accessdate=2013-12-17 }}&lt;/ref&gt;

: &lt;math&gt;EM = \int_0^\infty n_e^2 ds.&lt;/math&gt;

==Vectors==
{{main|Vectors}}
[[Image:3D Vector.svg|100px|thumb|right]] For standard basis, or unit, vectors ('''i''', '''j''', '''k''') there may be vector components of '''a''' ('''a'''&lt;sub&gt;x&lt;/sub&gt;, '''a'''&lt;sub&gt;y&lt;/sub&gt;, '''a'''&lt;sub&gt;z&lt;/sub&gt;).

'''Def.''' a directed quantity, one with both magnitude and direction; the signed difference between two points is called a '''vector'''.

&quot;An observed time series consists of ''N'' data values x(t&lt;sub&gt;α&lt;/sub&gt;) taken at a set of ''N'' discrete times {t&lt;sub&gt;α&lt;/sub&gt;}. Hence it defines an ''N''-dimensional ''contravariant vector'' in ''sampling space'', by taking as the α&lt;sup&gt;th&lt;/sup&gt; component of the vector, the value of the data at time t&lt;sub&gt;α&lt;/sub&gt;, i.e.,
:&lt;math&gt;x^\alpha = [x({t_1}),x({t_2}),...,x({t_N})].&lt;/math&gt;
This representation is the ''canonical basis'' for sampling space.&quot;&lt;ref name=Foster/&gt;
{{clear}}

==Tensors==
{{main|Tensors}}
'''Def.''' a mathematical object consisting of a set of components with n indices each of which range from 1 to m where n is the rank and m is the dimension is called a '''tensor'''.

&quot;An impressive array of time series analysis methods are equivalent to treating the data as a vector in function space, then projecting the data vector onto a subspace of low dimension. A geometric approach isolates and exposes many of the important features of time series techniques, directly adapts to irregular time spacing, and easily accommodates variable statistical weights. Tensor notation provides an ideal formalism for these techniques. It is quite convenient for distinguishing a variety of different vector spaces, and is the most compact notation for all the sums which arise in the analysis.&quot;&lt;ref name=Foster&gt;{{ cite journal
|author=Grant Foster
|title=Time Series Analysis by Projection. II. Tensor Methods for Time Series Analysis
|journal=The Astronomical Journal
|month=January
|year=1996
|volume=111
|issue=1
|pages=555-65
|url=http://adsabs.harvard.edu/full/1996AJ....111..555F
|arxiv=
|bibcode=1996AJ....111..555F
|doi=
|pmid=
|accessdate=2013-12-16 }}&lt;/ref&gt;

&quot;[T]he generally invariant line element

: &lt;math&gt; d s^2 = g_{\mu \nu} dx^{\mu}dx^{\nu} &lt;/math&gt;

[contains] the spacetime metric tensor &lt;math&gt;g_{\mu \nu} (x^{\rho}), \mu, \nu, \rho = 0, 1, 2, 3,&lt;/math&gt; [which] plays a dual role: on the one hand it determines the spacetime geometry, on the other it represents the (ten components of the) gravitational potential, and is thus a dynamical variable.&quot;&lt;ref name=Bicak&gt;{{ cite journal
|author=Jiří Bičák
|title=Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics, In: ''Einstein’s Field Equations and Their Physical Implications''
|journal=Lecture Notes in Physics
|publisher=Springer Berlin Heidelberg
|location=Berlin
|month=
|year=2000
|editor=
|volume=540
|issue=
|pages=1-126
|url=http://arxiv.org/pdf/gr-qc/0004016
|arxiv=
|bibcode=
|doi=10.1007/3-540-46580-4_1
|pmid=
|isbn=978-3-540-67073-5
|accessdate=2013-07-04 }}&lt;/ref&gt;

==Electronic computers==
{{main|Electronic computers}}
'''Def.''' a programmable electronic device that performs mathematical calculations and logical operations, especially one that can process, store and retrieve large amounts of data very quickly; now especially, a small one for personal or home use employed for manipulating text or graphics, accessing the Internet, or playing games or media is called a '''computer'''.

A '''computer''' is a general purpose device that can be [[w:Computer program|programmed]] to carry out a finite set of arithmetic or logical operations. Since a sequence of operations can be readily changed, the computer can solve more than one kind of problem.

==Programmings==
{{main|Programmings}}
A '''computer program''' (also '''[[w:computer software|software]]''', or just a '''program''') is a sequence of [[w:instruction (computer science)|instructions]] written to perform a specified task with a computer.&lt;ref name=&quot;pis-ch4-p132&quot;&gt;{{ cite book
  | last = Stair
  | first = Ralph M.
  |display-authors=etal
  | title = Principles of Information Systems, Sixth Edition
  | publisher = Thomson Learning, Inc.
  | date = 2003
  | pages = 132
  | isbn = 0-619-06489-7 }}&lt;/ref&gt; A computer requires programs to function, typically [[w:execution (computing)|executing]] the program's instructions in a [[w:central processing unit|central processor]].&lt;ref name=&quot;osc-ch3-p58&quot;&gt;{{ cite book
  | last = Silberschatz
  | first = Abraham
  | title = Operating System Concepts, Fourth Edition
  | publisher = Addison-Wesley
  | date = 1994
  | pages = 58
  | isbn = 0-201-50480-4 }}&lt;/ref&gt;

'''Computer programming''' (often shortened to '''programming''' or '''coding''') is the process of [[w:Software design|designing]], writing, [[w:Software testing|testing]], [[w:debugging|debugging]], and maintaining the [[w:source code|source code]] of [[w:computer program|computer program]]s.

==Probabilities==
{{main|Probabilities}}
'''Def.''' a number, between 0 and 1, expressing the precise likelihood of an event happening is called a '''probability'''.

'''Probability''' is a measure of the expectation that an event will occur or a statement is true. Probabilities are given a value between 0 (will not occur) and 1 (will occur).&lt;ref name=Feller&gt;{{ cite book
|author=William Feller
|date=1968
|title=An Introduction to Probability Theory and its Applications
|volume=1
|ISBN=0-471-25708-7
}}&lt;/ref&gt;  The higher the probability of an event, the more certain we are that the event will occur.

==Statistics==
{{main|Statistics}}
'''Def.''' a  mathematical science concerned with data collection, presentation, analysis, and interpretation is called '''statistics'''.

'''[[Statistics]]''' is the study of the collection, organization, analysis, interpretation, and presentation of [[w:data|data]].&lt;ref name=Dodge&gt;Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}}&lt;/ref&gt;&lt;ref&gt;[http://www.thefreedictionary.com/dict.asp?Word=statistics The Free Online Dictionary]&lt;/ref&gt; It deals with all aspects of this, including the planning of data collection in terms of the design of [[w:statistical survey|survey]]s and [[w:experimental design|experiments]].&lt;ref name=Dodge/&gt;

&quot;Statistics of projections are derived under a number of different null hypotheses.&quot;&lt;ref name=Foster/&gt;

==Hypotheses==
{{main|Hypotheses}}
# Each mathematical approach requires a proof of concept.

==See also==
{{div col|colwidth=12em}}
* [[w:Cosmic distance ladder|Cosmic distance ladder]]
* [[Portal:Euclidean geometry|Euclidean geometry]]
* [[Topic:Mathematical physics|Mathematical physics]]
* [[Portal:Mathematics|Mathematics]]
* [[Radiation astronomy/Courses/Principles|Principles of Radiation Astronomy]]
* [[Probability]]
* [[Radiation astronomy/Mathematics|Radiation mathematics]]
* [[Statistics]]
* [[Astronomy/Theory|Theoretical astronomy]]
* [[Radiation astronomy/Theory|Theoretical radiation astronomy]]
* [[Topic:Trigonometry|Trigonometry]]
{{Div col end}}

==References==
{{reflist|2}}

==Further reading==
* {{ cite book
|author=William Marshall Smart
|author2=Robin Michael Green
|title=Textbook on Spherical Astronomy, Sixth Edition
|publisher=University of Cambridge
|location=Cambridge
|date=July 7, 1977
|editor=
|pages=431
|url=http://books.google.com/books?id=W0f2vc2EePUC&amp;lr=&amp;source=gbs_navlinks_s
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=0 521 21516 1
|accessdate=2012-05-18 }}
* {{ cite journal 
| title=Large-scale expanding superstructures in galaxies 
| author=Tenorio-Tagle G
|author2=Bodenheimer P
| journal=Annual Review of Astronomy and Astrophysics 
| date=1988 
|volume=26 
|issue= 
|pages=145–97 
|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T 
}}

==External links==
* [http://www.bing.com/search?q=&amp;go=&amp;qs=n&amp;sk=&amp;sc=8-15&amp;qb=1&amp;FORM=AXRE Bing Advanced search]
* [http://books.google.com/ Google Books]
* [http://scholar.google.com/advanced_scholar_search?hl=en&amp;lr= Google scholar Advanced Scholar Search]
* [http://www.iau.org/ International Astronomical Union]
* [http://www.jstor.org/ JSTOR]
* [http://www.lycos.com/ Lycos search]
* [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED]
* [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center.]
* [http://www.questia.com/ Questia - The Online Library of Books and Journals]
* [http://online.sagepub.com/ SAGE journals online]
* [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System]
* [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search]
* [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer]
* [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database]
* [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA]
* [http://www.springerlink.com/ SpringerLink]
* [http://www.tandfonline.com/ Taylor &amp; Francis Online]
* [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter]
* [http://onlinelibrary.wiley.com/advanced/search Wiley Online Library Advanced Search]
* [http://search.yahoo.com/web/advanced Yahoo Advanced Web Search]

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== Reminder to vote now to select members of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
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This is a reminder that the voting period for the Universal Code of Conduct Coordinating Committee (U4C) ends on May 9, 2024. Read the information on the [[m:Universal Code of Conduct/Coordinating Committee/Election/2024|voting page on Meta-wiki]] to learn more about voting and voter eligibility.

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    <title>Understanding Arithmetic Circuits</title>
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      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|pdf]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|pdf]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|pdf]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|pdf]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20211111.pdf|pdf]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|pdf]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|pdf]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|pdf]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240502.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|pdf]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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    <title>Esperanto/Lesson 3</title>
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      <text bytes="13305" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Esperanto/Lesson 2|Previous lesson]] — [[Topic:Esperanto|Main page]] — [[Esperanto/Lesson 4|Next lesson]]
&lt;/p&gt;
We will learn about [[w:Adverb|adverbs]] in Esperanto, get to know the word &quot;ĉu&quot;, the suffixes &quot;-ulo&quot; and &quot;-ino&quot;, the ending &quot;-n&quot;, and learn the numbers in Esperanto.

==Adverbs==
Adverbs are words that usually describe a characteristic of a verb, adjective, another adverb, or an entire sentence. Examples can be found in the box directly below. 
{| class=&quot;wikitable collapsible collapsed&quot; border=&quot;1&quot; cellpadding=&quot;5&quot; 
|-style=&quot;background:#ffffaa&quot;
! Examples of adverbs in English
|-
| I '''quickly''' saw it.&lt;br&gt;He spoke '''quietly'''.&lt;br&gt;They '''often''' have new ideas.&lt;br&gt;I am '''quickly''' learning Esperanto '''well'''.&lt;br&gt;I have '''just''' started studying the new lesson.
|}

In Esperanto, adverbs are most often marked by the ending '''-e'''. These are adverbs derived from other words. We will discuss this derivation in the next lesson. Several common adverbs end in '''-aŭ''', which does not specify the part of speech of the word and is shared with some words that function as [[w:conjunction|conjunctions]] or prepositions, or do not have a specific ending. Examples are in the short vocabulary section below.

===Vocabulary===
Here are the first adverbs to memorize.
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Word !! Audio !! Meaning
|-
| laŭte || [[File:LL-Q143 (epo)-Lepticed7-laŭte.wav|frameless]]|| loudly
|-
| bone || [[File:Eo-bone.ogg|frameless]] || well
|-
| ĝuste || [[File:Eo-ĝuste.ogg|frameless]] || correctly
|-
| kutime || [[File:LL-Q143 (epo)-Robin van der Vliet-kutime.wav|frameless]]|| usually, ordinarily
|-
| ofte || [[File:Eo-ofte.ogg|frameless]] || often
|-
| ankaŭ || [[File:Eo-ankaŭ.ogg|frameless]] || also
|-
| baldaŭ || [[File:Eo-baldaŭ.ogg|frameless]] || soon
|-
| preskaŭ || [[File:Eo-preskaŭ.ogg|frameless]] || nearly, almost
|-
| nun || [[File:LL-Q143 (epo)-Lepticed7-nun.wav|frameless]]|| now
|-
| jam || [[File:Eo-jam.ogg|frameless]] || already
|-
| ja || [[File:LL-Q143 (epo)-Lepticed7-ja.wav|frameless]]|| indeed
|-
| jes || [[File:Eo-jes.ogg|frameless]] || yes
|-
| nur || [[File:Eo-nur.ogg|frameless]] || only, just
|-
| tuj || [[File:LL-Q143 (epo)-Lepticed7-tuj.wav|frameless]]|| immediately
|-
| ĵus || [[File:Eo-ĵus.ogg|frameless]] || just now
|}

===Placement===
Adverbs can be placed preceding or following the verb, adjective, or adverb that it modifies. Adverbs that modify an entire sentence typically come first. There is no difference in meaning. 

===Examples===
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Sentence !! Audio !! Meaning
|-
| La infano laŭte ploras. ||  || The child cries loudly.
|-
| Ŝi skribas bele. ||  || She writes beautifully.
|-
| Ili bone studas. ||  || They study well.
|-
| Mi venos baldaŭ. ||  || I will come soon.
|}

==ĉu==
To ask a yes/no question in Esperanto, one only has to place &quot;ĉu&quot; at the beginning of the sentence (and, of course, like in English, place a question mark at the end of a written sentence). Contrary to in English, word order does not change.

{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Sentence !! Audio !! Meaning
|-
| Ŝi volas promeni. — ''Ĉu'' ŝi volas promeni? ||  [[File:Eo-Ŝi volas promeni. — Ĉu ŝi volas promeni?.ogg|frameless]]|| She wants to walk. — Does she want to walk? 
|-
| Li estas malsana. — ''Ĉu'' li estas malsana? ||  [[File:Eo-Li estas malsana. — Ĉu li estas malsana?.ogg|frameless]]|| He is ill. — Is he ill?
|-
| La hundo estas en la ĝardeno. — ''Ĉu'' la hundo estas en la ĝardeno? ||  [[File:Eo-La hundo estas en la ĝardeno. — Ĉu la hundo estas en la ĝardeno?.ogg|frameless]]|| The dog is in the garden. — Is the dog in the garden?
|}


&quot;Ĉu&quot; can also be used in indirect questions, where it translates to 'whether'. It is then always preceded by a comma.

{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Sentence !! Audio !! Meaning
|-
| ''Ĉu'' li volas promeni? ||  [[File:Eo-Ĉu li volas promeni?.ogg|frameless]]|| Does he want to walk?
|-
|Mi ne scias, ''ĉu'' li volas promeni.
|[[File:Eo-Mi ne scias, ĉu li volas promeni.ogg|frameless]]
|I don't know ''whether'' he wants to walk (or not).
|-
| ''Ĉu'' li estas malsana? — Ŝi ne volas scii, ''ĉu'' li estas malsana. ||  [[File:Eo-Ĉu li estas malsana? — Ŝi ne volas scii, ĉu li estas malsana.ogg|frameless]]|| Is he ill? — She does not want to know ''whether'' he is ill (or not).
|-
| ''Ĉu'' la hundo estas en la ĝardeno? — ''Ĉu'' vi scias, ''ĉu'' la hundo estas en la ĝardeno? ||  [[File:Eo-Ĉu la hundo estas en la ĝardeno? — Ĉu vi scias, ĉu la hundo estas en la ĝardeno?.ogg|frameless]]|| Is the dog in the garden? — Do you know ''whether'' the dog is in the garden (or not)? 
|}

==-n==
The subject of a sentence is the 'doer' of the verb and the object is the 'undergoer' of the verb. Examples can be found in the box directly below. 
{| class=&quot;wikitable collapsible collapsed&quot; border=&quot;1&quot; cellpadding=&quot;5&quot; 
|-style=&quot;background:#ffffaa&quot;
! Examples of subjects and objects in English
|-
| &lt;small&gt;''subjects'' in italics; '''objects''' in bold&lt;/small&gt;
|-
| ''He'' saw '''me'''.&lt;br&gt;''He'' told '''a story''' to me.&lt;br&gt;''They'' have '''a new idea'''.&lt;br&gt;''I'' am learning '''Esperanto''' well.&lt;br&gt;''I'' am studying '''the new lesson'''.
|}

In English, what is the object and the subject of a sentence is indicated by where the word is in the sentence. As can be seen in the examples above, the subject (in italics) comes before the verb and the object (in bold) comes after the verb. Many pronouns in English have distinct subject and object forms: I vs. me, he vs. him, she vs. her, etc..

In Esperanto, the '''object''' of a sentence is indicated by adding the ending '''-n''' to nouns and accompanying adjectives (which may already have taken the plural ending -j) or pronouns, ''not'' by the order of the words in the sentence. The article (&quot;la&quot;) does not take ''-n''. Even though word order does not matter in Esperanto, there is a default word order, which is the same as the normal word order in English (first the subject, then the verb, then the object).

===Examples===
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Sentence !! Audio !! Meaning
|-
| Mi vidas la hundon.&lt;br&gt;La hundon mi vidas.&lt;br&gt;La hundon vidas mi. ||  [[File:Eo-Mi vidas la hundon. La hundon mi vidas. La hundon vidas mi.ogg|frameless]]|| I see the dog.
|-
| Li ne vidas la grandajn hundojn, nur la malgrandajn (hundojn). ||  [[File:Eo-Li ne vidas la grandajn hundojn, nur la malgrandajn (hundojn).ogg|frameless]]|| He does not see the big dogs, just the small ones/dogs.
|-
| La patrino amas la filinon.&lt;br&gt;La filinon la patrino amas.&lt;br&gt;La filinon amas la patrino. ||  [[File:Eo-La patrino amas la filinon. La filinon la patrino amas. La filinon amas la patrino.ogg|frameless]]|| The mother loves the daughter.
|-
| Ŝi ne aŭdis min.&lt;br&gt;Min ŝi ne aŭdis.&lt;br&gt;Min ne aŭdis ŝi. || [[File:Eo-Ŝi ne aŭdis min. Min ŝi ne aŭdis. Min ne aŭdis ŝi.ogg|frameless]] || She did not hear me.
|-
| Kutime mi ne komprenas la lecionojn.&lt;br&gt;La lecionojn kutime mi ne komprenas. ||[[File:Eo-Kutime mi ne komprenas la lecionojn. La lecionojn kutime mi ne komprenas.ogg|frameless]]  || I usually don't understand the lessons.
|-
| Mi ĵus vidis la fortan viron. ||  [[File:Eo-Mi ĵus vidis la fortan viron.ogg|frameless]]|| I have seen the strong man just now.
|}

==The numbers==
The basic numbers in Esperanto are: 
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Numeral !! Word !! Audio
|-
| 1 || unu || [[File:Eo-unu.ogg|frameless]]
|-
| 2 || du || [[File:Eo-du.ogg|frameless]]
|-
| 3 || tri || [[File:Eo-tri.oga|frameless]]
|-
| 4 || kvar || [[File:Eo-kvar.ogg|frameless]]
|-
| 5 || kvin || [[File:Eo-kvin.ogg|frameless]]
|-
| 6 || ses || [[File:Eo-ses.ogg|frameless]]
|-
| 7 || sep || [[File:Eo-sep.ogg|frameless]]
|-
| 8 || ok || [[File:Eo-ok.ogg|frameless]]
|-
| 9 || naŭ || [[File:Eo-naŭ.ogg|frameless]]
|-
| 10 || dek || [[File:Eo-dek.ogg|frameless]]
|-
| 100 || cent || [[File:Eo-cent.ogg|frameless]]
|-
| 1000 || mil || [[File:Eo-mil.ogg|frameless]]
|}

These regularly combine as follows:
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Numeral !! Word !! Audio !! Numeral !! Word !! Audio
|-
| 20 || dudek || [[File:Eo-dudek.ogg|frameless]] || 11 || dek unu || [[File:LL-Q143 (epo)-Lepticed7-dek unu.wav|frameless]]
|-
| 30 || tridek || [[File:Eo-tridek.ogg|frameless]] || 12 || dek du || [[File:LL-Q143 (epo)-Lepticed7-dek du.wav|frameless]]
|-
| 40 || kvardek || [[File:Eo-kvardek.ogg|frameless]] || 13 || dek tri || [[File:LL-Q143 (epo)-Lepticed7-dek tri.wav|frameless]]
|-style=text-align:center
| colspan=3 | etc. || colspan=3 | etc. 
|}

==Affixes==
===-ulo===
The suffix &quot;-ulo&quot; indicates a word that refers to someone characterized by the base word.
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! colspan=5 | Examples 
|-style=&quot;background:lime&quot;
! Base word !! Meaning !! Derived word !! Meaning !! Audio
|-
| griza || gray || grizulo || graybeard, gray-haired person || 
|-
| malsana || ill, sick || malsanulo || sick person || 
|-
| blonda || blond || blondulo || blond person || 
|-
| kara || lovely, dear || karulo || darling ||
|-
| maljuna || old || maljunulo || old person ||
|}

===-ino===
'''-ino''' is a suffix to indicate female sex.
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! colspan=5 | Examples 
|-style=&quot;background:lime&quot;
! Base word !! Meaning !! Derived word !! Meaning !! Audio
|-
| viro || man || virino || woman || 
|-
| amiko || friend || amikino || female friend || 
|-
| hundo || dog || hundino || female dog, bitch ||
|-
| patro || father || patrino || mother || 
|-
| frato || brother || fratino || sister || 
|-
| knabo || boy || knabino || girl || 
|-
| blondulo || blond || blondulino || blond female person || 
|}

==Vocabulary==
Here are more words to memorize.
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Word !! Audio !! Meaning
|-
| havi ||  || to have
|-
| vidi ||  || to see
|-
| kuraci || || to cure
|-
| aŭdi ||  || to hear
|-
| legi ||  || to read
|-
| danci ||  || to dance
|-
| forgesi ||  || to forget
|-
| dormi ||  || to sleep
|-
| veni ||  || to come
|-
| stari ||  || to stand
|-
| loĝi ||  || to live, to reside
|-
| hieraŭ ||  || yesterday
|-
| hodiaŭ ||  || today
|-
| morgaŭ ||  || tomorrow
|-
| ŝuo ||  || shoe
|-
| helpi ||  || to help
|-
| porti ||  || to carry
|-
| mordi ||  || to bite
|-
| ĥoro ||  || choir
|-
| aĉeti ||  || to buy
|-
| vendi ||  || to sell
|-
| butiko ||  || shop
|-
| aperta ||  || open
|-
| kvankam ||  || although
|-
| ..., do ... ||  || ..., so ...
|-
| botelo ||  || bottle
|-
| ..., ke ... ||  || ... that ...
|-
| aperi ||  || to appear 
|-
| ankoraŭ ||  || still
|-
| ĉar ||  || because
|-
| plori ||  || to cry
|-
| teo ||  || tea
|-
| kafo ||  || coffee
|-
| pensi ||  || to think
|-
| nubo ||  || cloud
|}

==Exercises==
{{expand section|date=December 2012}}
Write out the following numbers: ([[/Answers#Numbers|answers]])
* 11, 14, 21, 34, 77, 17, 99, 67, 76, 54, 38, 22, 83, 92, 50, 93, 87, 78, 90
* 108, 104, 112, 184, 200, 307, 503, 808, 818, 311, 271, 511, 401, 837, 983, 543, 651, 765, 345, 813, 173, 148, 713, 607, 670, 573, 963, 842, 633, 752, 937
* 2014, 2007, 1954, 1945, 1859, 1914, 1801, 1777, 1653, 2373, 8472, 2367, 2151, 2263, 2177, 8371, 8297, 7939, 4567, 7654, 9369, 3713, 5407, 4095

Now, let's play with affixes:
*Go through the words in this lesson and tack on ''mal-''. What do they mean then? ([[/Answers#mal-|answers]])
*Go through the words in this lesson and the previous one and tack on ''-ulo'' and ''-ino'' wherever meaningful. Also see if you can tack on both and either or both with ''mal-''. What do these words mean then? ([[/Answers#-ulo and -ino|answers]])

There are one or more grammatical mistakes in each of the following five sentences. Where?
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Sentence &lt;small&gt;([[/Answers#Correct them|answers]])&lt;/small&gt;
|-
| Mi ne volas vidi li.
|-
| Ŝi ne aŭdis la granda ĥoron.
|-
| La hundo mordos la viro.
|-
| Hieraŭ ni iros al lin.
|-
| Ni portis la malsana knabo al la kuracisto. 
|}

You can continue practicing by translating the following sentences:
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Sentence &lt;small&gt;([[/Answers#Esperanto–English|answers]])&lt;/small&gt; !! Audio
|-
| La fratino estas bela. || 
|-
| Li amas knabinojn. || 
|-
| Hodiaŭ la bela virino venos al ni, sed la malbela kato ne venos. || 
|-
| Ŝi forgesis, ke hieraŭ vi trinkis la akvon de la hundo! || 
|-
| Ĉu vi volas trinki teon aŭ kafon? || 
|}

{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Sentence &lt;small&gt;([[/Answers#English–Esperanto|answers]])&lt;/small&gt;
|-
| I see two clouds in the sky.
|- 
|I went across the river.
|}

&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Esperanto/Lesson 2|Previous lesson]] — [[Topic:Esperanto|Main page]] — [[Esperanto/Lesson 4|Next lesson]]
&lt;/p&gt;

[[Category:Esperanto]]</text>
      <sha1>nr5m5mnq25flvhpzudfdlmi4tkll0tf</sha1>
    </revision>
  </page>
  <page>
    <title>Esperanto/Root chart</title>
    <ns>0</ns>
    <id>147668</id>
    <revision>
      <id>2624600</id>
      <parentid>2536180</parentid>
      <timestamp>2024-05-02T14:07:43Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10823" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-ar-o-j ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gust
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/fold || du-obl-a ||  || double
|-
| -on ||  || part || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair
|-
| -ul ||  || trait || stult-ul-o ||  || fool
|-
| -uj ||  || container || Mon-uj-o ||  || purse/wallet
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:bluish|bluish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellowish|yellowish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:grayish|grayish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orangy|orangy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purpley|purpley]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pinkish|pinkish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:redish|redish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlety|scarlety]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoisy|turquoisy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:greenish|greenish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>guw4qtpfuzimt6459kvzgezd0xx268h</sha1>
    </revision>
    <revision>
      <id>2624602</id>
      <parentid>2624600</parentid>
      <timestamp>2024-05-02T14:09:27Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <minor />
      <comment>/* Suffixes */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10834" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-ar-o-j ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gust
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/fold || du-obl-a ||  || double
|-
| -on ||  || part || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair
|-
| -ul ||  || trait || stult-ul-o ||  || fool
|-
| -uj ||  || container || Mon-uj-o ||  || purse/wallet
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:bluish|bluish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellowish|yellowish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:grayish|grayish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orangy|orangy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purpley|purpley]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pinkish|pinkish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:redish|redish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlety|scarlety]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoisy|turquoisy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:greenish|greenish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>orrmyijm1pn2fs2s9f8kwo87a61u5i0</sha1>
    </revision>
    <revision>
      <id>2624604</id>
      <parentid>2624602</parentid>
      <timestamp>2024-05-02T14:12:24Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <comment>/* Suffixes */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10857" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-an-ar-o ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gale
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || to become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/-fold || du-obl-a ||  || double
|-
| -on ||  || part, fraction || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair, couple
|-
| -ul ||  || trait || stult-ul-o ||  || fool
|-
| -uj ||  || container || mon-uj-o ||  || purse/wallet
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:bluish|bluish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellowish|yellowish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:grayish|grayish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orangy|orangy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purpley|purpley]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pinkish|pinkish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:redish|redish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlety|scarlety]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoisy|turquoisy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:greenish|greenish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>6u53uas8ockxknnhvci3cl6s2xpzm1h</sha1>
    </revision>
    <revision>
      <id>2624605</id>
      <parentid>2624604</parentid>
      <timestamp>2024-05-02T14:13:27Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <comment>/* Suffixes */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10860" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-an-ar-o ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gale
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || to become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-an-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/-fold || du-obl-a ||  || double
|-
| -on ||  || part, fraction || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair, couple
|-
| -ul ||  || trait || stult-ul-o ||  || fool
|-
| -uj ||  || container || mon-uj-o ||  || purse/wallet
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:bluish|bluish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellowish|yellowish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:grayish|grayish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orangy|orangy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purpley|purpley]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pinkish|pinkish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:redish|redish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlety|scarlety]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoisy|turquoisy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:greenish|greenish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>oip9gv0omzdebuf9z2pvpxfagvb1arc</sha1>
    </revision>
    <revision>
      <id>2624606</id>
      <parentid>2624605</parentid>
      <timestamp>2024-05-02T14:15:37Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <comment>/* Suffixes */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10870" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-an-ar-o ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gale
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || to become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-an-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/-fold || du-obl-a ||  || double
|-
| -on ||  || part, fraction || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair, couple
|-
| -uj ||  || container || mon-uj-o ||  || purse/wallet
|-
| -ul ||  || person of trait || stult-ul-o ||  || fool
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:bluish|bluish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellowish|yellowish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:grayish|grayish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orangy|orangy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purpley|purpley]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pinkish|pinkish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:redish|redish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlety|scarlety]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoisy|turquoisy]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:greenish|greenish]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>7kjaiom88aguc5ny6o1y8ry92a29vab</sha1>
    </revision>
    <revision>
      <id>2624621</id>
      <parentid>2624606</parentid>
      <timestamp>2024-05-02T14:27:38Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <comment>/* Colors */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10832" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;typically used as a third-gender pronoun and is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-an-ar-o ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gale
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || to become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-an-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/-fold || du-obl-a ||  || double
|-
| -on ||  || part, fraction || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair, couple
|-
| -uj ||  || container || mon-uj-o ||  || purse/wallet
|-
| -ul ||  || person of trait || stult-ul-o ||  || fool
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:blue|blue]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellow|yellow]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:gray|gray]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orange|orange]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purple|purple]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pink|pink]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:red|red]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlet|scarlet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoise|turquoise]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:green|green]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>mjnwb1tjzizanaxfwn0dac0ip8yooq6</sha1>
    </revision>
    <revision>
      <id>2624624</id>
      <parentid>2624621</parentid>
      <timestamp>2024-05-02T14:29:54Z</timestamp>
      <contributor>
        <username>Sdiabhon Sdiamhon</username>
        <id>2947922</id>
      </contributor>
      <comment>/* Personal Pronouns */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10854" xml:space="preserve">&lt;p style=&quot;background: #f2f2f2; border: 1px dashed #e6e6e6; padding-left: 3px; width: 100%; text-align: center;&quot;&gt;
[[Topic:Esperanto|Main page]]
&lt;/p&gt;
This is an attempt to create a convenient list of the roots and their meanings in English

Proper nouns have been omitted. - and should not be changed.

This has gotten too large for wiki to save it efficiently in its entirety so it has now been moved [http://home.ceneezer.com/esperanto/ to my site] where you can view in a more customized way, and can still contribute to it if you choose to login (ctrl+right-click).

==Personal Pronouns==
{| class=wikitable
|-
! colspan=2 | !! singular !! plural
|-
! colspan=2 | first person
| '''mi''' (I) || '''ni''' (we)
|-
! colspan=2 | second person
| colspan=2 align=center | '''vi''' (you)
|-
! rowspan=&quot;4&quot; | third&lt;br&gt;person !! masculine
| '''li''' (he) || rowspan=&quot;4&quot; | '''ili''' (they)
|-
! feminine
| '''ŝi''' (she)
|-
! [[Wiktionary:epicene|epicene]]
| '''ĝi''' (it)
|-
!neutral
|'''ri*''' (they)
|}

&lt;nowiki&gt;*&lt;/nowiki&gt;nonstandard, typically used as a third-gender pronoun and using it is called [[wikipedia:Ri_(pronoun)|''riismo'']] ''(they-ism)''

==Correlatives==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:lime&quot;
! Correlative Table !! K- !! T- !! (alone) !! Ĉ- !! Nen- || al-
(non-standard &amp; likely to cause confusion)
|-
! -u&lt;br&gt;(Unique person/thing) || k-i-u&lt;br&gt;(who) || t-i-u&lt;br&gt;(this/that person/thing) || i-u&lt;br&gt;(some/a person/thing) || ĉ-i-u&lt;br&gt;(every person/thing) || nen-i-u&lt;br&gt;(No person/thing) || al-i-u&lt;br&gt;(someone else)
|-
! -e&lt;br&gt;(place, location) || k-i-e&lt;br&gt;(where) || t-i-e&lt;br&gt;(there) || i-e&lt;br&gt;(some place) || ĉ-i-e&lt;br&gt;(every place) || nen-i-e&lt;br&gt;(No place) || al-i-e&lt;br&gt;(elsewhere)
|-
! -o&lt;br&gt;(abstract thing) || k-i-o&lt;br&gt;(what) || t-i-o&lt;br&gt;(that) || i-o&lt;br&gt;(some thing) || ĉ-i-o&lt;br&gt;(every thing) || nen-i-o&lt;br&gt;(No thing) || al-i-o&lt;br&gt;(something else)
|-
! -a&lt;br&gt;(type/kind) || k-i-a&lt;br&gt;(what type, which kind) || t-i-a&lt;br&gt;(this/that type/kind) || i-a&lt;br&gt;(some/a type/kind) || ĉ-i-a&lt;br&gt;(every type/kind) || nen-i-a&lt;br&gt;(No type/kind) || al-i-a&lt;br&gt;(another kind)
|-
! -om&lt;br&gt;(quantity) || k-i-om&lt;br&gt;(how many) || t-i-om&lt;br&gt;(this many) || i-om&lt;br&gt;(some amount) || ĉ-i-om&lt;br&gt;(every amount) || nen-i-om&lt;br&gt;(No amount) || al-i-om&lt;br&gt;(another amount)
|-
! -am&lt;br&gt;(time) || k-i-am&lt;br&gt;(when) || t-i-am&lt;br&gt;(then) || i-am&lt;br&gt;(some time) || ĉ-i-am&lt;br&gt;(every time, always) || nen-i-am&lt;br&gt;(No time, never) || al-i-am&lt;br&gt;(another time)
|-
! -es&lt;br&gt;(ownership) || k-i-es&lt;br&gt;(whose) || t-i-es&lt;br&gt;(their) || i-es&lt;br&gt;(some/any one's) || ĉ-i-es&lt;br&gt;(everyone's) || neni-es&lt;br&gt;(noone's) || al-i-es&lt;br&gt;(someone else's)
|-
! -al&lt;br&gt;(causation) || k-i-al&lt;br&gt;(why) || t-i-al&lt;br&gt;(because) || i-al&lt;br&gt;(for some reason) || ĉ-i-al&lt;br&gt;(for all reasons) || nen-i-al&lt;br&gt;(for no reason) || al-i-al&lt;br&gt;(another reason)
|-
! -el&lt;br&gt;(method) || k-i-el&lt;br&gt;(how, in what method) || t-i-el&lt;br&gt;(in this method) || i-el&lt;br&gt;(in some method) || ĉ-i-el&lt;br&gt;(in every way) || nen-i-el&lt;br&gt;(in no way) || al-i-el&lt;br&gt;(another way)
|}


==Prepositions==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! preposition !! Meaning
|-
| al || to, toward
|-
| anstataŭ || instead of
|-
| antaŭ || before, in front of
|-
| apud || beside, near, nearby, near to, next to
|-
| ĉe || at (generic non-specific preposition)
|-
| ĉirkaŭ || about, around, circa
|-
| de ||  of, from, by
|-
| dum || during, for, while
|-
| ekde || since, starting/beginning at
|-
| el || from, out of
|-
| en || into, inside, inside of
|-
| ekster || outside, outside of
|-
| eksteren || outward, towards the outside of
|-
| ĝis || until, till
|-
| inter || among, between
|-
| kontraŭ || against, across from, in exchange for, opposed to, opposite
|-
| krom || except for, except, apart from, besides
|-
| kun || with, together with
|-
| laŭ || according to
|-
| malgraŭ || despite, in spite of
|-
| ol || than (more/less)
|-
| per || by means of, through use of
|-
| plus || numerically greater than
|-
| post || after, behind
|-
| po || at the rate of
|-
| por || on behalf of
|-
| preter || beyond
|-
| pri || about, concerning, on, upon
|-
| pro || because of
|-
| sen || without
|-
| sub || below, beneath
|-
| sur || upon, on top of
|-
| super || above, over
|-
| tra || through (as in a location, not through use/means of)
|-
| trans || across, beyond, on the other side of
|}
&lt;br&gt;

==Suffixes==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:palegreen&quot;
! Suffix !! Audio !! Meaning !! Example !! Audio !! Translation
|-
| -aĉ ||  || cheap, poor quality || dom-aĉ-o ||  || shack
|-
| -ad ||  || prolonged || parol-ad-i ||  || ramble
|-
| -aĵo ||  || to make a noun || sek-aĵ-o ||  || dry good
|-
| -an ||  || citizen, member of || Kanad-an-o ||  || Canadian
|-
| -ar ||  || group || Kanad-an-ar-o ||  || Canadians
|-
| -ĉj ||  || male nickname || pa-ĉj-o ||  || daddy
|-
| -ebl ||  || possibility || vid-ebl-a ||  || visible
|-
| -ec ||  || -ness || dolĉ-ec-o ||  || sweetness
|-
| -eg ||  || bigger || vent-eg-o ||  || gale
|-
| -ej ||  || to make a place || preĝ-ej-o ||  || church/temple (place of prayer)
|-
| -em ||  || tendency || ŝpar-em-a ||  || thrifty
|-
| -end ||  || should/must || leg-end-a ||  || that should be read
|-
| -er ||  || part of || sal-er-o ||  || grain of salt
|-
| -estr ||  || leader || Kanad-estr-o ||  || Prime-minister (Canadian leader)
|-
| -et ||  || smaller || rid-et-i ||  || smile (little laugh)
|-
| -id ||  || offspring || ĉeval-id-o ||  || colt
|-
| -ig ||  || to cause || pur-ig-i ||  || to make clean
|-
| -iĝ ||  || to become || ruĝ-iĝ-i ||  || blush (redden)
|-
| -il ||  || tool || ŝlos-il-o ||  || key (lock tool)
|-
| -in ||  || female || Kanad-an-in-o ||  || Canadian woman
|-
| -ind ||  || worthy || admir-ind-a ||  || admirable
|-
| -ist ||  || profession || Esperant-ist-o ||  || Esperanto teacher/speaker
|-
| -nj ||  || female nickname || pa-nj-o ||  || mommy
|-
| -obl ||  || times/-fold || du-obl-a ||  || double
|-
| -on ||  || part, fraction || kvar-on-o ||  || quarter
|-
| #op ||  || group || du-op-o ||  || pair, couple
|-
| -uj ||  || container || mon-uj-o ||  || purse/wallet
|-
| -ul ||  || person of trait || stult-ul-o ||  || fool
|}

==Colors==
{| border=&quot;1&quot; cellpadding=&quot;5&quot;
|-style=&quot;background:red&quot;
! root !! Akademio de Esperanto !! -o&lt;br&gt;(noun) !! -a&lt;br&gt;(adjective) !! -i&lt;br&gt;(verb) !! -e&lt;br&gt;(adverb) !! mal-&lt;br&gt;(opposite) !! chef-&lt;br&gt;(main) !! ek-&lt;br&gt;(start) !! eks&lt;br&gt;(end) !! re-&lt;br&gt;(cyclic) !! dis-&lt;br&gt;(distributive) !! pra-&lt;br&gt;(ancient) !! comments
|-
|[[w:eo:arĝent|arĝent]]||silver||[[w:silver|silver]] || [[w:silvery|silvery]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blank|blank]]||white||[[w:white|white]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blond|blond]]||fair||[[w:blond|blond]] || [[w:blond|blond]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:blu|blu]]||blue||[[w:blue|blue]] || [[w:blue|blue]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:brun|brun]]||brown||[[w:brown|brown]] || [[w:brown|brown]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ebon|ebon]]||ebony||[[w:ebony|ebony]] || [[w:ebony|ebony]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:flav|flav]]||yellow||[[w:yellow|yellow]] || [[w:yellow|yellow]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:griz|griz]]||grey||[[w:gray|gray]] || [[w:gray|gray]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kobalt|kobalt]]||cobalt||[[w:Cobalt|Cobalt]] || [[w:cobalty|cobalty]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:kolor|kolor]]||color||[[w:color|color]] || [[w:colorfull|colorful]] || [[w:color|color]] || [[w:|w:]] || [[w:monochome|monochome]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:nigr|nigr]]||black||[[w:black|black]] || [[w:dark|dark]] || [[w:|w:]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:oranĝ|oranĝ]]||orange||[[w:orange|orange]] || [[w:orange|orange]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:pal|pal]]||pale||[[w:|w:]] || [[w:pale|pale]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:purpur|purpur]]||purple||[[w:purple|purple]] || [[w:purple|purple]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:roz|roz]]||rose||[[w:pink|pink]] || [[w:pink|pink]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:ruĝ|ruĝ]]||red||[[w:red|red]] || [[w:red|red]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:skarlat|skarlat]]||scarlet||[[w:scarlet|scarlet]] || [[w:scarlet|scarlet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:stri|stri]]||stripe/streak||[[w:strip|strip]] || [[w:|w:]] || [[w:stripe|stripe]] || [[w:|w:]] || [[w:solid|solid]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:tan|tan]]||tan||[[w:tan|tan]] || [[w:|w:]] || [[w:tan|tan]] || [[w:|w:]] || [[w:light|light]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:turkis|turkis]]||turquoise||[[w:turquoise|turquoise]] || [[w:turquoise|turquoise]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:verd|verd]]||green||[[w:green|green]] || [[w:green|green]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|[[w:eo:viol|viol]]||violet||[[w:violet|violet]] || [[w:violet|violet]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]] || [[w:|w:]]||	
|-
|}

{{Hitcounter}}

{{CourseCat}}</text>
      <sha1>8kbfxs5cx8a60pv7toeyvma963m8bau</sha1>
    </revision>
  </page>
  <page>
    <title>User:Guy vandegrift/Editorial tricks</title>
    <ns>2</ns>
    <id>157776</id>
    <revision>
      <id>2624878</id>
      <parentid>2616311</parentid>
      <timestamp>2024-05-02T23:56:25Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18590" xml:space="preserve">{{#timel: l d F Y g:i:A|+6 hours}} {{Purge|refresh}}&lt;code&gt;&lt;nowiki&gt; {{Purge|refresh}}&lt;/nowiki&gt;&lt;/code&gt;  
{{search box |root=User:Guy vandegrift/Editorial tricks |width=100|break=no|label=Search this page}}
*&lt;nowiki&gt;[[Special:Diff/583498451/611653557|&lt;text&gt;]]&lt;/nowiki&gt;
*[[QB]] -- [[Help:Interwiki linking]] -- [[w:Template:Graph:Chart]] -- [[w:Help:Special_page#Available_special_pages]]
*[https://www.tablesgenerator.com/mediawiki_tables Tables Generator] -- [https://tools.wmflabs.org/excel2wiki/ Copy and Paste Excel-to-Wiki Converter]]
*The script for a redirect is &lt;code&gt;&lt;nowiki&gt;#REDIRECT [[&lt;text&gt;]]&lt;/nowiki&gt;&lt;/code&gt; where &lt;text&gt; is a pagename.
*[[w:Template:CSS image crop]]
*&lt;code&gt;&lt;nowiki&gt;{{RoundBoxTop}} ...{{RoundBoxBottom}}&lt;/nowiki&gt;&lt;/code&gt; See also {{tl|Template:Robelbox}} '''&lt;code&gt;&lt;nowiki&gt;{{tl|Template:Robelbox}}&lt;/nowiki&gt;&lt;/code&gt;'''
*&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;thumb tleft&quot;&gt;&lt;div style=&quot;width:100px;&quot;&gt;xx&lt;/div&gt;&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;
*&lt;nowiki&gt;{{BASEPAGENAME}}&lt;/nowiki&gt; creates {{BASEPAGENAME}}
*&lt;nowiki&gt;Use {{colbegin}} and {{colend}} templates to produce columns.&lt;/nowiki&gt; See also [[Template:Multicol]]
----
{{User:Guy_vandegrift/T/Usertitle}}
{{User:Guy vandegrift/T/Projects}} [https://ftools.toolforge.org/ ftools.toolforge.org] 
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&lt;div style=&quot;font-size:75%; line-height: 1em&quot;;&gt; small &lt;/div&gt;
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&lt;nowiki&gt;{{REVISIONID}}&lt;/nowiki&gt;={{REVISIONID}} .. Titleparts:{{Titleparts|1}} . {{Titleparts|2}} . {{Titleparts|3}}

&lt;nowiki&gt;{{TOC limit|2}}  {{Citation needed}}   {{tl2|Cite journal}}&lt;/nowiki&gt;

*[[mw:Help:Magic_words]]999
*&lt;nowiki&gt;&amp;lt;syntaxhighlight lang='matlab'&amp;gt;&lt;/nowiki&gt;

better way to break: &lt;nowiki&gt;&lt;wbr&gt;&amp;nbsp;&lt;/nowiki&gt; (contains nbsp)

See [[w:Wikipedia:Advanced text formatting]] and [[w:User:Guy vandegrift/sandbox]] (w:Editorial tricks)

&lt;pre&gt;International Phonetic Alphabet - SIL
International Phonetic Alphabet - X-SAMPA
Use native keyboardCTRL+M ...Which do I want?&lt;/pre&gt;
====Download====
([[file:OOjs UI icon download.svg|18px]]&lt;span class=&quot;plainlinks&quot;&gt;[https://en.wikiversity.org/w/index.php?title=Special:ElectronPdf&amp;page={{Space_to_underscore|{{FULLPAGENAME}}}} Download]&lt;/span&gt;)
 
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[[:Category:Navigational templates]] | [[Template:EasyNav]] | [[Template:Draftspace]]

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&lt;math&gt;QWERTYUIOP\;ASDFGHJKL \;ZXCVBNM&lt;/math&gt;

&lt;math&gt;qwertyuiop\;asdfghjkl\;zxcvbnm&lt;/math&gt;

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38-Newfunction: row1: {&lt;!--c24ElectromagneticWaves_displacementCurrent_2--&gt;A circlular capactitor of radius  3.2 m has a gap of 13 mm, and a charge of 49 &amp;mu;C.  Compute the surface integral  &lt;math&gt;c^{-2}\oint\vec E\cdot d\vec A&lt;/math&gt; over an inner face of the capacitor.}
38-Newfunction: row3: {&lt;!--AstroApparentRetroMotion_7--&gt; If a planet that is very, very far from the Sun begins a retrograde, how many months must pass before it begins the next retrograde?  }
38-Newfunction: row4: {&lt;!--AstroGalileanMoons_5--&gt;Immediately after publication of Newton's laws of physics (Principia), it was possible to &quot;calculate&quot; the mass of Jupiter.  What important caveat applied to this calculation?   }
38-Newfunction: row5: {&lt;!--AstroLunarphasesAdvancedB_53--&gt;At 6pm a waning crescent moon would be} 
38-Newfunction: row6: {&lt;!--c22Magnetism_ampereLawSymmetry_3--&gt;H is defined by, B=&amp;mu;&lt;sub&gt;0&lt;/sub&gt;H, where B is magnetic field. A current of 84A passes along the z-axis.  Use symmetry to find the integral, &lt;math&gt;\int \vec H\cdot\vec{d\ell}&lt;/math&gt;, from the point {{nowrap begin}}(0,9.3){{nowrap end}} to the point {{nowrap begin}}(9.3,9.3){{nowrap end}}.}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. Curabitur pretium tincidunt lacus. Nulla gravida orci a odio. Nullam varius, turpis et commodo pharetra, est eros bibendum elit, nec luctus magna felis sollicitudin mauris. Integer in mauris eu nibh euismod gravida. Duis ac tellus et risus vulputate vehicula. Donec lobortis risus a elit. Etiam tempor. Ut ullamcorper, ligula eu tempor congue, eros est euismod turpis, id tincidunt 
&lt;/nowiki&gt;
&lt;/pre&gt;

====Textfiles====

fprintf(fout,'%s\n','text');
====Centered and right margined text====
&lt;pre&gt;
center:
&lt;div class=&quot;center&quot; style=&quot;width: auto; margin-left: auto; margin-right: auto;&quot;&gt;text&lt;/div&gt;
center small:
&lt;div class=&quot;center&quot; style=&quot;width: auto; margin-left: auto; margin-right: auto;&quot;&gt;text&lt;/div&gt;
right:
&lt;div style=&quot;text-align: right; direction: ltr; margin-left: 1em;&quot;&gt;text&lt;/div&gt; 
right small:
&lt;div style=&quot;text-align: right; direction: ltr; margin-left: 1em;&quot;&gt;text&lt;/div&gt; 

&lt;/pre&gt; 
*[[w:Help:Wiki_markup#Center_text|Center text]]
*[[Template:Blockquote]]

====Converting equation into image format====
See image and caption to the right: The first six rows of Pascal's triangle
&lt;div class=&quot;thumb tright&quot;&gt;
&lt;div style=&quot;width:500px;&quot;&gt;
&lt;math&gt;
\begin{array}{lc}
(a+b)^0= &amp;
{\color{Red}\boldsymbol{1}} 
\\
(a+b)^1=  &amp; 
{\color{Red}\boldsymbol{1}}a+{\color{Red}\boldsymbol{1}}b
\\
(a+b)^2=  &amp;
{\color{Red}\boldsymbol{1}}a^2+{\color{Red}\boldsymbol{2}}ab+{\color{Red}\boldsymbol{1}}b^2
\\
(a+b)^3=  &amp;
{\color{Red}\boldsymbol{1}}a^3+{\color{Red}\boldsymbol{3}}a^2b+{\color{Red}\boldsymbol{3}}ab^2+{\color{Red}\boldsymbol{1}}b^3
\\
(a+b)^4=  &amp;
{\color{Red}\boldsymbol{1}}a^4+{\color{Red}\boldsymbol{4}}a^3b+{\color{Red}\boldsymbol{6}}a^2b^2+{\color{Red}\boldsymbol{4}}ab^3+ {\color{Red}\boldsymbol{1}}b^4
\\                                     
\end{array}
&lt;/math&gt;
&lt;div class=&quot;thumbcaption&quot;&gt;
The first six rows of Pascal's triangle&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;

====Transclusion====
See '''[[#Labeled and unlabeled transclusion|below]]''' for actual templates for copy/paste
*[[w:Wikipedia:Transclusion#Selective_transclusion template transclusion]] How to transclude a labeled section from a template or another website.
*[[w:Help:Labeled_section_transclusion|Labeled section transclusion]]
**[[mw:Extension:Labeled_Section_Transclusion|Labeled Section Transclusion (wikimedia.org)]]

====Import requests====
*[[Wikiversity:Import| Import request]]

===Symbols and colors===
*&amp;‌#2384; &amp;‌#x0950; ॐ &lt;span class=&quot;Unicode&quot;&gt;U+0950 &lt;/span&gt;
* A&amp;#8407;B
*[[w:User:Guy_vandegrift/sandbox#Symbols]]
* html tags such as &amp;infin; &amp;alpha; &amp;#8467; &amp;frac12; &amp;asymp; &amp;plusmn;(&amp;infin &amp;alpha &amp;#8467 &amp;frac12 &amp;asymp &amp;plusmn)... can be found at 
**http://www.freeformatter.com/html-entities.html
**http://www.dionysia.org/html/entities/symbols.html
**http://www.w3schools.com/charsets/ref_html_entities_4.asp
* html tags: http://www.w3schools.com/charsets/ref_utf_math.asp
* [http://www.rpi.edu/dept/arc/training/latex/LaTeX_symbols.pdf Great Big List of Latex Symbols]
* [[wikipedia:Miscellaneous_Symbols|Wikitext Symbols]]:Sun☉ Star☆ Moon☽ Earth♁ⴲ [[w:Astronomical_symbols#Symbols_for_the_planets|planets]]
*[[w:List_of_colors_(compact)]]

*http://www.javascripter.net/faq/mathsymbols.htm &lt;code&gt;&amp;deg;&lt;/code&gt;

===Tags: http://www.w3schools.com/charsets/default.asp&lt;nowiki/&gt;===
*[[metametawiki:Help:HTML in wikitext]]

== Equations ==

=== numbered equations ===
{{NumBlk|:|&lt;math&gt;
 E=mc^2         
 &lt;/math&gt;|{{EquationRef|G5}}}} 

{{EquationRef|G5}} 

{{NumBlk|:|a|b}}

&lt;cite&gt;h&lt;/cite&gt;
*In [[Divergence theorem]] I used [[Template:NumBlk]] and referenced using '''Equation{{spaces|1}}5''':

{{NumBlk|2=&lt;math&gt;\mathbf{\vec F}=F_x(x)\mathbf\hat i + F_y(x)\mathbf\hat j&lt;/math&gt;
|3=5}}

*In [[Poynting's theorem]]  I used [[Template:NumBlk]]:

{{NumBlk|:|&lt;math&gt;dV\rightarrow dxdydz \leftrightarrow d\tau \leftrightarrow d^3x \leftrightarrow d\mathcal{V}_\text{ol}&lt;/math&gt;|1|LnSty=1px dashed}}

==== Two templates I might like ====
{{See|w:Template:EquationRef|w:Template:EquationNote}}

==== Possible problem ====
In [[Rule_of_product]] I used {{EquationRef|Figure 3}} as the link and {{EquationNote|Figure 2}}as the target.  The templates are [[Template:EquationRef]] as link [[Template:EquationNote]] as target.

'''No space is allowed after EquationNote:''' &lt;nowiki&gt;{{EquationNote|Figure 2}}as the target.&lt;/nowiki&gt; yields {{EquationNote|Figure 2}}as the target.

*In [[Draft:Information theory/Permutations and combinations]] I place equation ref in front: {{EquationRef|Eq. 2}}{{spaces|2}}&lt;math&gt;\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{3!}{2!1!}&lt;/math&gt; and refer using the internal link: bla bla {{EquationNote|Fig. 5}}to arbitrary values on {{math|n}} and {{math|k}} as as stated at {{EquationNote|Eq. 2}}.On the other hand, an important restriction results from using a three-dimensional box to account for '''''duplicate''''' words
 [[Template:EquationNote]] is '''not advised''' due to unexpected results. In the above example the * (Bullet list) symbol forces the pagebreak.

=== Multiple equations&lt;ref&gt;https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Multiple_equations&lt;/ref&gt; ===
&lt;math&gt;\begin{matrix}
u &amp; = \tfrac{1}{\sqrt{2}}(x+y) \qquad &amp; x &amp;= \tfrac{1}{\sqrt{2}}(u+v)\\
v &amp; = \tfrac{1}{\sqrt{2}}(x-y) \qquad &amp; y &amp;= \tfrac{1}{\sqrt{2}}(u-v)
\end{matrix}&lt;/math&gt;
&amp;nbsp;
&lt;syntaxhighlight lang=&quot;text&quot;&gt;
&lt;math&gt;\begin{align}
11 &amp; = 12 &amp; 13  \\
21 &amp; = 22 &amp; y 23
\end{align}&lt;/math&gt;
&lt;/syntaxhighlight&gt;

==== trial ====
&lt;math&gt;\left[\begin{array}{c|c}
     x -2\sigma         &amp;        \sigma           &amp;         0   \\

      2\sigma         &amp;       -2\sigma-\beta\epsilon        &amp;     \beta\epsilon 
\end{array}\right]&lt;/math&gt;

==== arrays1 ====
&lt;math&gt;\left[\begin{array}{c|c}
     x -2\sigma         &amp;        \sigma           &amp;         0   \\
\hline
      2\sigma         &amp;       -2\sigma-\beta\epsilon        &amp;     \beta\epsilon    \\
\hline
     0             &amp;        \epsilon            &amp;  x -\alpha   -\epsilon   
\end{array}\right]\cdot\left[\begin{array}{c|c}
     y_1   \\
\hline
      y_2   \\
\hline
     y_A  
\end{array}\right] = 0
&lt;/math&gt;

==== arrays2 ====
&lt;math&gt;\begin{pmatrix}
     x -2\sigma         &amp;        \sigma           &amp;         0   \\

      2\sigma         &amp;       -2\sigma-\beta\epsilon        &amp;     \beta\epsilon    \\

     0             &amp;        \epsilon            &amp;  x -\alpha   -\epsilon   
\end{pmatrix}\cdot\begin{bmatrix}
     y_1   \\

      y_2   \\

     y_A  
\end{bmatrix}\ = 0
&lt;/math&gt;

==Labeled and unlabeled transclusion==
===Transclusion of unlabeled sections===
&lt;nowiki&gt;{{#lsth:OpenStax College|See_Also}}&lt;/nowiki&gt;
{{cot|click to view transclusion}}
{{#lsth:OpenStax College|See Also}}
{{cob}}

===Labeled section transclusion===
See also [[w:Help:Labeled_section_transclusion]].
How to label
:&lt;nowiki&gt;&lt;section begin=chapter1 /&gt;this is a chapter&lt;section end=chapter1 /&gt;&lt;/nowiki&gt;
How to call the text
:&lt;nowiki&gt;{{#lst:resource_page_name/subpage|Title_of_section}}&lt;/nowiki&gt;  
{{cot|click to view transclusion}} 
{{#lst:Physics_equations/Oscillations,_waves,_and_interference|sho_energy}}
{{cob}}

===comment===
To understand these examples, visit:
*[[Physics_equations/Oscillations,_waves,_and_interference]]
*[[#lst:Physics_equations/Oscillations,_waves,_and_interference]]
===Please contact me (about quizzes) -- [[Quizbank/How to use testbank|edit message]]===
&lt;nowiki&gt;{{#lsth:Quizbank/How to use testbank|Please contact me}}&lt;/nowiki&gt;

==Box answers==
&lt;math&gt; \underline{\overline{ \left | ANSWER\right |}}&lt;/math&gt;
----
&lt;nowiki&gt;\underline{\overline{ \left | ANSWER\right |}}&lt;/nowiki&gt;
----
===quote box===
[[w:template:Quote box]]

{{quote box
|halign=left
|align=left
|quote= 
&lt;math&gt;P(t)=I(t)V(t) = I_\text{rms}V_\text{rms}\cos(\phi_v-\phi_i) + I_\text{rms}V_\text{rms}\cos(2\omega t+\phi_v+\phi_i) &lt;/math&gt;
}}{{clear}}

===box===
{| class=&quot;toccolours&quot; style=&quot;float: left; margin-left: 1em; margin-right: 2em; font-size: 95%; background:#c6dbf7; color:black; width:30em; max-width: 40%;&quot; cellspacing=&quot;5&quot;
| style=&quot;text-align: left;&quot; |First line.
::Indented 
|}
{{clear}}
{| class=&quot;toccolours&quot; style=&quot;float: right; margin-left: 1em; margin-right: 2em; font-size: 95%; background:#c6dbf7; color:black; width:30em; max-width: 40%;&quot; cellspacing=&quot;5&quot;
| style=&quot;text-align: left;&quot; |First line.
::Indented 
|}

==Hidden text==
http://www.newtonproject.sussex.ac.uk/prism.php?id=40
===Comment out text for editors===
&lt;nowiki&gt;&lt;!-- Hidden text --&gt;&lt;/nowiki&gt;

===Grey bar white lightgrey(hides tables)===
{{hidden begin
|title = click to view or hide|
|titlestyle=background:lightgrey;}}
*foo
{{hidden end}}

===Grey bar white not lightgrey(also hides tables)===
{{hidden begin|title = click to view or hide}}
*foo
{{hidden end}}

===two colors 75% collapsible===
&lt;!--BEGIN HIDDEN TEXT (do not remove this comment--&gt;
:{| class=&quot;toccolours collapsible collapsed&quot; width=&quot;75%&quot; style=&quot;text-align:left&quot;
! To see WHATEVER click to right ===&gt;
|- &lt;!--KEEP THIS LINE UNTOUCHED; next line begins with &quot;|&quot;. --&gt;
| {{Lorem ipsum}}
|}&lt;!--KEEP THIS LINE UNTOUCHED &quot;|}&quot; unhides--&gt;
&lt;!--END HIDDEN TEXT (do not remove this comment--&gt;

===Jascript hidden text===

{{Noprint|{{JavaScript required|'''Enable JavaScript to hide answers.'''}}}}
{{Noprint|'''Click on a question to see the answer.'''}}
{{Collapsible toggle|collapsed=true|style=margin-bottom:0.5em;|toggle=1 | 1.
Visible1
|hidden1
}}
{{Collapsible toggle|collapsed=true|style=margin-bottom:0.5em;|toggle=1 | 2.
Visible2
|hidden2
}}

==page breaks==
See also [[w:Template:Page break]]
===Insert page break===
&lt;pre&gt;
&lt;div style=&quot;page-break-before:always&quot;&gt;&lt;/div&gt;&lt;!--simple pagebreak--&gt;
&lt;/pre&gt;

===Keep together===
&lt;pre&gt;
&lt;div style=&quot;page-break-inside:avoid;&quot;&gt;&lt;!--next section--&gt;
text
text
&lt;/div&gt;&lt;!--keep together--&gt;
&lt;/pre&gt;

==Footnotes and references==
&lt;nowiki&gt;&lt;ref group=&quot;note&quot;&gt;This goes into a footnote section.&lt;/ref&gt;&lt;/nowiki&gt;

Call with &lt;nowiki&gt;{{reflist|group=&quot;note&quot;|liststyle=lower-alpha}}&lt;/nowiki&gt;

==Subpages==
&lt;pre&gt;
&lt;nowiki&gt;
{{cot|list of subpages}}{{Subpages/List}}{{cob}} -- this one lists them all
{{Subpages/Simple}} -- links with single word &quot;Subpages&quot;
{{Subpages}} -- Links with [[List]] of subpages.
&lt;/nowiki&gt;

&lt;/pre&gt;

==Images==
===pixel counts and upright factors===
Although pixel counts are easier to understand than upright factors, they adjust less well to user preferences. For example, suppose a picture contains some detail and by default is a bit too small, and you want to grow it by about 10%. Although &quot;&lt;code&gt;upright=1.1&lt;/code&gt;&quot; and &quot;&lt;code&gt;240px&lt;/code&gt;&quot; do the job equally well for the common case where the default width is 220 pixels, many of the users who set the default width to 300 pixels to work better with their high-resolution screens will be annoyed with &quot;&lt;code&gt;200px&lt;/code&gt;&quot; because it will make the picture a third smaller than their preferred size. In contrast, &quot;&lt;code&gt;upright=1.1&lt;/code&gt;&quot; will display the picture to them with a width of 330 pixels, and this is more likely to work well on their displays.

Pixel counts are typically better than upright factors for displaying combinations of pictures, some of which have known and limited sizes, and for displaying tiny icons that are intended to be combined with text.

==Active user template==
The user inserts &lt;nowiki&gt;{{subst:Contrib-using}}&lt;/nowiki&gt; if  teaching.

The user inserts &lt;nowiki&gt;{{subst:Contrib-creator}}&lt;/nowiki&gt; if constructing.

See also [[Template:Please don't edit]]
==Sister interlinks==
&lt;div class=&quot;noprint infobox sisterproject&quot; style=&quot;border: solid #aaa 1px; clear: right; margin: 0 0 1em 1em; font-size: 90%; background: #f9f9f9; width: 250px; padding: 4px; spacing: 0px; text-align: left; float: right;&quot;&gt;
&lt;div style=&quot;float: left;&quot;&gt;[[Image:Wikiversity-logo-fr-pure.svg|50px]]&lt;/div&gt;
&lt;div style=&quot;margin-left: 55px;&quot;&gt;First line 
&lt;div style=&quot;margin-left: 10px;&quot;&gt;second line&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;!--begin textbox QUIZBANK for OPENSTAX COLLEGE--&gt;
&lt;div class=&quot;noprint infobox sisterproject&quot; style=&quot;border: solid #aaa 1px; clear: right; margin: 0 0 1em 1em; font-size: 90%; background: #f9f9f9; width: 250px; padding: 4px; spacing: 0px; text-align: left; float: right;&quot;&gt;&lt;div style=&quot;float: left;&quot;&gt;[[Image:Wikiversity-logo-fr-pure.svg|50px]]&lt;/div&gt;
&lt;div style=&quot;margin-left: 55px;&quot;&gt; '''[[Quizbank]]''' under construction &lt;div style=&quot;margin-left: 10px;&quot;&gt; for  '''[[OpenStax College]]'''&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;
&lt;!--end textbox QUIZBANK for OPENSTAX COLLEGE--&gt;


===Interlink prefixes===
*[[w:Main page]] or [[Wikipedia:Main page]]
*[[v:Main page]] but not [[Wikiversity:Main page]]
*[[mw:Main page]] 
*[[v:Main page]] or [[Wikibooks:Main page]]
*[[wikt:Main page]] or [[Wiktionary:Main page]]

==side by side image==
{{multiple image
| align     = left
| image1    = Right hand rule cross product large print.svg
| width1    = {{#expr: (100 * 750 /400) round 0}}
| caption1  = [[w:simple:special:permalink/6003961#Visualizing_the_cross_product_in_three_dimensions|Cross product]] &lt;math&gt;|\vec a\times\vec b|=ab\sin\theta&lt;/math&gt;
| image2    = Hall effect for OpenStax Physics formula sheet.svg
| width2    = 150
| caption2  =
}}
===multiple images===
&lt;nowiki&gt;{| style=&quot;border-spacing: 1px; align:center&quot;
|-
|[[File:Entropy_flip_2_coins.jpg|thumb|160px|'''Figure 1'''. Entropy is, &lt;br&gt;&lt;math&gt;S=2=N&lt;/math&gt;, &lt;br&gt;where &lt;math&gt;N&lt;/math&gt; is the number of fair coins. The number of possible messages is: &lt;br&gt;&lt;math&gt;\Omega=2^N= 4&lt;/math&gt;. ]]
|[[File:Shannon entropy 5 coin illustration.svg |'''Figure 2'''. Elephant with 5 bits of entropy can give 3 bits to Bird and 2 bits to Rat.|thumb|210px]]
|[[File:Shannon entropy coins and base 2.svg|Figure 3|thumb|270px]]
|- 
|}&lt;/nowiki&gt;

==pdf license for quizbank and openstax==
{{clear}}
==Annotated image==
[[Template:Annotated image]]

== Licensing ==
{{cc-by-3.0}}
[[Category:openstax file]]
[[Category:WSUL files]]
[[Category:Quizbank]]

==Python code==
‎&lt;syntaxhighlight lang=&quot;python&quot; line&gt;
def quick_sort(arr):
	less = []
	pivot_list = []
	more = []
	if len(arr) &lt;= 1:
		return arr
	else:
		pass
‎&lt;/syntaxhighlight&gt;


‎&lt;syntaxhighlight&gt;
def quick_sort(arr):
	less = []
	pivot_list = []
	more = []
	if len(arr) &lt;= 1:
		return arr
	else:
		pass
‎&lt;/syntaxhighlight&gt;

==MyOpenMath==
*Quizbank physics 1 60675
*Quizbank physics 2 
&lt;!--guy_vandegrift Guy Vandegrift
sock_vandegrift--sockOfGuy Vandegrift&gt;</text>
      <sha1>3bkdpceucsa4s0wakknmdxrxmfw5w6w</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2624607</id>
      <parentid>2624431</parentid>
      <timestamp>2024-05-02T14:19:57Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5186" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20240501.pdf|B.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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      <comment>/* Geometric Series Examples */</comment>
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      <text bytes="5186" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20240502.pdf|B.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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{{Information
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{{Information
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*[[Wikiversity Journal of Medicine/Estimating the lost benefits of visual inspection with acetic acid for cervical cancer prevention in South Africa|Estimating the lost benefits of visual inspection with acetic acid for cervical cancer prevention in South Africa]]
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{{Information
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&lt;br&gt;Comments on the peer review are located at the [[Talk:Wikiversity Journal of Medicine/Estimating the lost benefits of not implementing a visual inspection with acetic acid screen and treat strategy for cervical cancer prevention in South Africa|Talk page of that article]]
|Source         =Own work
|Author         =Usha Rani Poli. &lt;br&gt;Uploaded by [[User:Mikael Häggström|Mikael Häggström]]
|Date           =2015-08-16
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{{self|GFDL|cc-by-sa-3.0}}

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{{Information
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|Source=[[Wikiversity Journal of Medicine/Estimating the lost benefits of not implementing a visual inspection with acetic acid screen and treat strategy for cervical cancer prevention in South Africa]]
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{{Information
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|Source={{cite journal|last1=Meyers|first1=Craig|last2=Parham|first2=Groesbeck P.|last3=Mwanahamuntu|first3=Mulindi H.|last4=Kapambwe|first4=Sharon|last5=Muwonge|first5=Richard|last6=Bateman|first6=Allen C.|last7=Blevins|first7=Meridith|last8=Chibwesha|first8=Carla J.|last9=Pfaendler|first9=Krista S.|last10=Mudenda|first10=Victor|last11=Shibemba|first11=Aaron L.|last12=Chisele|first12=Samson|last13=Mkumba|first13=Gracilia|last14=Vwalika|first14=Bellington|last15=Hicks|first15=Michael L.|last16=Vermund|first16=Sten H.|last17=Chi|first17=Benjamin H.|last18=Stringer|first18=Jeffrey S. A.|last19=Sankaranarayanan|first19=Rengaswamy|last20=Sahasrabuddhe|first20=Vikrant V.|title=Population-Level Scale-Up of Cervical Cancer Prevention Services in a Low-Resource Setting: Development, Implementation, and Evaluation of the Cervical Cancer Prevention Program in Zambia|journal=PLOS ONE|volume=10|issue=4|year=2015|pages=e0122169|issn=1932-6203|doi=10.1371/journal.pone.0122169}}
|Date=2015
|Author=Meyers, Craig; Parham, Groesbeck P.; Mwanahamuntu, Mulindi H.; Kapambwe, Sharon; Muwonge, Richard; Bateman, Allen C.; Blevins, Meridith; Chibwesha, Carla J. et al.
|Permission=
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&lt;!-- 在本行下編輯模板說明 --&gt;
== About ==
'''&lt;nowiki&gt;{{font}}&lt;/nowiki&gt;''' This template is used to define text styles, for example, size and color. It can also be used to inject [[CSS]]. This is very useful when it is needed to change &quot;I&quot; to &quot;{{font|I|font=Times}}&quot;, since uppercase &quot;I&quot; looks the same as lowercase &quot;L&quot;.

== How to ==
&lt;code&gt;&lt;nowiki&gt;{{font|TEXT(or「text=TEXT」)|font=FONT|size=SIZE（px/em/pt/%）|color=COLOR|bgcolor=BACKGROUND COLOR|css=CSS}}&lt;/nowiki&gt;&lt;/code&gt;

== Example ==
If enter &lt;code&gt;&lt;nowiki&gt;{{font|text=Some text.|font=Comic Sans MS|size=20px|color=#7f5620}}&lt;/nowiki&gt;&lt;/code&gt;&lt;br&gt;
It will show as: {{font|text=Some text.|font=Comic Sans MS|size=20px|color=#7f5620}}

If enter &lt;code&gt;&lt;nowiki&gt;{{font|Illinois|font=Times|size=20px}}&lt;/nowiki&gt;&lt;/code&gt;&lt;br&gt;
It will show as: {{font|Illinois|font=Times|size=20px}}, which is way better than {{font|Illinois|size=20px}}.

{| class=&quot;wikitable&quot;
!Enter
!Will show as
|-
|&lt;code&gt;&lt;nowiki&gt;{{font|啡色的文字。|font=標楷體|size=20px|color=#7f5620}}&lt;/nowiki&gt;&lt;/code&gt;
|{{font|啡色的文字。|font=標楷體|size=20px|color=#7f5620}}
|-
|&lt;code&gt;&lt;nowiki&gt;{{font|text=Hello World!|font=Century Gothic|size=35px|color=#bf00bf}}&lt;/nowiki&gt;&lt;/code&gt;
|{{font|text=Hello World!|font=Century Gothic|size=35px|color=#bf00bf}}
|-
|&lt;code&gt;&lt;nowiki&gt;{{font|text=囧囧囧囧囧|font=Simhei|color=#c9b295|bgcolor=#364d6a}}&lt;/nowiki&gt;&lt;/code&gt;
|{{font|text=囧囧囧囧囧|font=Simhei|color=#c9b295|bgcolor=#364d6a}}
|-
|&lt;code&gt;&lt;nowiki&gt;{{font|text=可以選用全部選項。|font=Simhei|size=20px|color=#5a7aad|bgcolor=#f9f9ef}}&lt;/nowiki&gt;&lt;/code&gt;
|{{font|text=可以選用全部選項，|font=Simhei|size=20px|color=#5a7aad|bgcolor=#f9f9ef}}
|-
|&lt;code&gt;&lt;nowiki&gt;{{font|text=也可以只選用其中一項選項。|size=25px}}&lt;/nowiki&gt;&lt;/code&gt;
|{{font|text=也可以只選用其中一項選項。|size=25px}}
|}

== Attention ==
* By default:
{| class=&quot;WSPP KING&quot;
!Parameter
!Default
|-
|&lt;code&gt;size&lt;/code&gt;
|100%
|-
|&lt;code&gt;color&lt;/code&gt;
|black
|-
|&lt;code&gt;bgcolor&lt;/code&gt;
|transparent
|}

== See also ==
* {{tl|font color}}
* {{tl|font-family georgia}}

&lt;includeonly&gt;
[[Category:Text color templates]]
[[Category:Font templates]]

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      <model>wikitext</model>
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      <text bytes="4761" xml:space="preserve">{{languages}}
&lt;div style=&quot;color: green; font-size: 200%;&quot;&gt;{{center|'''Bonvenon!'''}}&lt;/div&gt;
{{center|[[File:Flag of Esperanto.svg‎|95px]]}}

Welcome to the '''Department of Esperanto''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language Studies|School of Language Studies]].

== Introduction ==

Esperanto is a constructed language that was published in 1887 by {{w|L. L. Zamenhof}} with the intention for it to become the language of international communication. Although it has not yet achieved this ambitious goal, Esperanto is by far the most successful constructed language. It is much easier to learn than any typical natural language and learning it has a {{w|Propaedeutic value of Esperanto|positive effect}} on learning more languages. This course aims to take the learner from the basics through all the intricacies of the language.

== Course ==

&lt;div style=&quot;float: right; margin: 2px;&quot;&gt;
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
|{{yawn}}
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Vocabulary|Vocabulary]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Grammar|Grammar]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Grammar Rules|16 Grammar Rules]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Idioms|Idioms]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Irregularities|Irregularities]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[Esperanto/Root chart|Root chart]]
|-
|{{center|'''Additional Wikimedia resources'''}}
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:b:Esperanto|Textbook]] at Wikibooks
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | {{w|Esperanto|Article}} at Wikipedia
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | {{wikt|Index:Esperanto|Word index}} at the English Wiktionary
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[wikt:eo:Ĉefpaĝo|The Esperanto Wiktionary]]
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:s:The Esperanto Teacher|The Esperanto Teacher]] at Wikisource
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:s:Dr. Esperanto's International Language|Dr. Esperanto's International Language]] at Wikisource
|}&lt;/div&gt;

* [[Esperanto/Lesson 1|Lesson 1]] {{100Percent}} — Pronunciation, parts of speech, noun basics, article, a few personal pronouns, present tense
* [[Esperanto/Lesson 2|Lesson 2]] {{100Percent}} — Pronunciation (repetition), adjective basics, verb basics, personal pronouns, introduction to affixes (''mal-'')
* [[Esperanto/Lesson 3|Lesson 3]] {{75Percent}} — Adverbs, ''ĉu'', the suffixes ''-ulo'' and ''-ino'', the ending ''-n'', and counting
* [[Esperanto/Lesson 4|Lesson 4]] {{75Percent}} — Basic derivation
* [[Esperanto/Lesson 5|Lesson 5]] {{50Percent}} — Imperatives, introduction to correlatives
* [[Esperanto/Lesson 6|Lesson 6]] {{50Percent}} — Derivation with prepositions; Theme: Locations (incl. ''-ejo'' and ''-ujo''), directions, and movement (incl. ''dis-'')
* [[Esperanto/Lesson 7|Lesson 7]] {{50Percent}} — Theme: Family (incl. ''bo-'', ''ge-'', ''pra-'', ''vic-'', and ''eks-'')
* [[Esperanto/Lesson 8|Lesson 8]] {{50Percent}} — Derivation using numerals
* [[Esperanto/Lesson 9|Lesson 9]] {{25Percent}} — Advanced derivation; Theme: Time
* [[Esperanto/Lesson 10|Lesson 10]] {{0Percent}} — Theme: Food
* [[Esperanto/Lesson 11|Lesson 11]] {{0Percent}} — Theme: Home
* [[Esperanto/Lesson 12|Lesson 12]] {{0Percent}} — Theme: The body
* [[Esperanto/Lesson 13|Lesson 13]] {{25Percent}} — Theme: The history of Esperanto
* [[Esperanto/Lesson 14|Lesson 14]] {{0Percent}} — Theme: Communication
* [[Esperanto/Lesson 15|Lesson 15]] {{0Percent}} — Theme: Weather and geography
* [[Esperanto/Lesson 16|Lesson 16]] {{0Percent}} — Theme: Work and occupations?

== External links ==
* {{URL|1=www.newenglishreview.org/custpage.cfm/frm/9560/sec_id/9560 |2=Why Esperanto is different}}
* {{URL|1=web.archive.org/web/20160316124912/http://donh.best.vwh.net/Esperanto/eaccess/eaccess.organizations.html |2=A list of Esperanto organizations}} (archived version)

== Participants ==

{{Main|Esperanto/Contributors}}

* [[User:Ceneezer|Ceneezer]] ([[User talk:Ceneezer|discuss]] • [[Special:Contributions/Ceneezer|contribs]])
* [[User:JorisvS|JorisvS]] ([[User talk:JorisvS|discuss]] • [[Special:Contributions/JorisvS|contribs]])

== Department news ==

* '''15 January 2014''' — First lesson fully completed.
* '''21 December 2012''' — First lesson launched!
* '''3 December, 2011''' — This page is being cleaned up.
* '''11 December, 2006''' — Department founded!

==See also==
* [[Introductory Esperanto]]

[[Category:Esperanto| ]]

[[el:Τμήμα:Εσπεράντο]]
[[es:Curso de Esperanto]]
[[fr:Département:Espéranto]]</text>
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&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
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Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
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==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
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[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
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====Gallery====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
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===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
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[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
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[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
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====Gallery====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
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===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
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[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
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==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
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&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
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===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function : &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 

x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
{{RoundBoxTop|theme=2}}
If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
{{RoundBoxBottom}}
===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>a6sb6xdxuo4legs8b97y9yeu2wb035w</sha1>
    </revision>
    <revision>
      <id>2624581</id>
      <parentid>2624579</parentid>
      <timestamp>2024-05-02T12:10:48Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Quadratic function */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="62402" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
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====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
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&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
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===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
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If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
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===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
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C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>hp65fkj3fzn5cc9us54jnttnjnwkter</sha1>
    </revision>
    <revision>
      <id>2624587</id>
      <parentid>2624581</parentid>
      <timestamp>2024-05-02T12:50:59Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Quadratic function */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="62543" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
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===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
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[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
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====Gallery====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
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===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
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==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
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==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
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===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
{{RoundBoxTop|theme=2}}
If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
{{RoundBoxBottom}}
===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

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====Welcomee====
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[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

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=====Welcomen=====
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     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>j9ug6c5bf9phn16m45gksod4yev3yvv</sha1>
    </revision>
    <revision>
      <id>2624589</id>
      <parentid>2624587</parentid>
      <timestamp>2024-05-02T13:28:12Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Quadratic function */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="64176" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
{{RoundBoxTop|theme=2}}
If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
{{RoundBoxBottom}}
===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

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====Welcomee====
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[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

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=====Welcomen=====
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     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>fn9kjk9sf1jvb1ucz8loq9prv81k6oa</sha1>
    </revision>
    <revision>
      <id>2624595</id>
      <parentid>2624589</parentid>
      <timestamp>2024-05-02T13:54:21Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* x = f(y) */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="64403" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
{{RoundBoxTop|theme=2}}
If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
{{RoundBoxBottom}}
===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

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====Welcomee====
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[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

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=====Welcomen=====
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     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>j1bapxp7uzm0ynarfj3rm1f8xy6ftfw</sha1>
    </revision>
    <revision>
      <id>2624601</id>
      <parentid>2624595</parentid>
      <timestamp>2024-05-02T14:07:44Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* x = f(y) */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="64528" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}

===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
{{RoundBoxBottom}}
==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
{{RoundBoxBottom}}
==Implementation==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==More calculations==
{{RoundBoxTop|theme=2}}
The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
{{RoundBoxBottom}}
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
{{RoundBoxTop|theme=2}}
If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
{{RoundBoxBottom}}
===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
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C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

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====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>fdota4ntyebmu0ap30312qr01pxkxxk</sha1>
    </revision>
    <revision>
      <id>2624871</id>
      <parentid>2624601</parentid>
      <timestamp>2024-05-02T23:51:02Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* x = f(y) */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="64529" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = 39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = 3799&lt;/math&gt;

&lt;math&gt;x^3 - 3xy^2 = -39582&lt;/math&gt;

&lt;math&gt;3x^2y - y^3 = -3799&lt;/math&gt;


=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Deriving the equation==
The curve is defined as a point whose distance to the focus and distance to a line, the directrix,
have a fixed ratio, eccentricity &lt;math&gt;e.&lt;/math&gt; Distance from focus to directrix must be non-zero.

Let the point have coordinates &lt;math&gt;(x,y).&lt;/math&gt;

Let the focus have coordinates &lt;math&gt;(p,q).&lt;/math&gt;

Let the directrix have equation &lt;math&gt;ax + by + c = 0&lt;/math&gt; where &lt;math&gt;a^2 + b^2 = 1.&lt;/math&gt;

Then &lt;math&gt;e = \frac {\text{distance to focus}}{\text{distance to directrix}}&lt;/math&gt; &lt;math&gt;= \frac{\sqrt{(x-p)^2 + (y-q)^2}}{ax + by + c}&lt;/math&gt;

&lt;math&gt;e(ax + by + c) = \sqrt{(x-p)^2 + (y-q)^2}&lt;/math&gt;

Square both sides: &lt;math&gt;(ax + by + c)(ax + by + c)e^2 = (x-p)^2 + (y-q)^2&lt;/math&gt;

Rearrange: &lt;math&gt;(x-p)^2 + (y-q)^2 - (ax + by + c)(ax + by + c)e^2 = 0\ \dots\ (1).&lt;/math&gt;

Expand &lt;math&gt;(1),&lt;/math&gt; simplify, gather like terms and result is:

&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0&lt;/math&gt; where:

&lt;math&gt;X = e^2&lt;/math&gt;

&lt;math&gt;A = Xa^2 - 1&lt;/math&gt;

&lt;math&gt;B = Xb^2 - 1&lt;/math&gt;

&lt;math&gt;C = 2Xab&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc&lt;/math&gt;

&lt;math&gt;F = Xc^2 - p^2 - q^2&lt;/math&gt;
{{RoundBoxTop|theme=2}}
Note that values &lt;math&gt;A,B,C,D,E,F&lt;/math&gt; depend on:

* &lt;math&gt;e&lt;/math&gt; non-zero. This method is not suitable for circle where &lt;math&gt;e = 0.&lt;/math&gt;

* &lt;math&gt;e^2.&lt;/math&gt; Sign of &lt;math&gt;e \pm&lt;/math&gt; is not significant.

* &lt;math&gt;(ax + by + c)^2.\ ((-a)x + (-b)y + (-c))^2&lt;/math&gt; or &lt;math&gt;((-1)(ax + by + c))^2&lt;/math&gt; and &lt;math&gt;(ax + by + c)^2&lt;/math&gt; produce same result.

For example, directrix &lt;math&gt;0.6x - 0.8y + 3 = 0&lt;/math&gt; and directrix &lt;math&gt;-0.6x + 0.8y - 3 = 0&lt;/math&gt;
produce same result.
{{RoundBoxBottom}}

==Implementation==

&lt;syntaxhighlight lang=python&gt;
# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 22


def reduce_Decimal_number(number) :
# This function improves appearance of numbers.
# The technique used here is to perform the calculations using precision of 22,
# then convert to float or int to display result.
# -1e-22 becomes 0.
#  12.34999999999999999999 becomes 12.35
# -1.000000000000000000001 becomes -1.
# 1E+1 becomes 10.
# 0.3333333333333333333333 is unchanged.
#
    thisName = 'reduce_Decimal_number(number) :'
    if type(number) != dD : number = dD(str(number))

    f1 = float(number)
    if (f1 + 1) == 1 : return dD(0)
    if int(f1) == f1 : return dD(int(f1))
        
    dD1 = dD(str(f1))

    t1 = dD1.normalize().as_tuple()
    if (len(t1[1]) &lt; 12) :
        # if number == 12.34999999999999999999, dD1 = 12.35
        return dD1

    return number


def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq, {}) :'.format(bool(flag))
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]

    divider = a**2 + b**2
    if divider == 0 :
        print (thisName, 'At least one of (a,b) must be non-zero.')
        return None
    if divider != 1 :
        root = divider.sqrt()
        a,b,c = [ (v/root) for v in (a,b,c) ]

    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None

    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q

    A,B,C,D,E,F = [ reduce_Decimal_number(v) for v in (A,B,C,D,E,F) ]

    if flag :
        print (thisName)
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (' ', str1)
        
    return (A,B,C,D,E,F)
&lt;/syntaxhighlight&gt;

==Examples==
===Parabola===
Every parabola has eccentricity &lt;math&gt;e = 1.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:0323parabola01.png|thumb|400px|'''Quadratic function complies with definition of parabola.'''
&lt;/br&gt;
Distance from point &lt;math&gt;(6,9)&lt;/math&gt; to focus = distance from point &lt;math&gt;(6,9)&lt;/math&gt; to directrix = 10.&lt;/br&gt;
Distance from point &lt;math&gt;(0,0)&lt;/math&gt; to focus = distance from point &lt;math&gt;(0,0)&lt;/math&gt; to directrix = 1.&lt;/br&gt;
]]
Simple quadratic function:

Let focus be point &lt;math&gt;(0,1).&lt;/math&gt;

Let directrix have equation: &lt;math&gt;y = -1&lt;/math&gt; or &lt;math&gt;(0)x + (1)y + 1 = 0.&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))
&lt;/syntaxhighlight&gt;
As conic section curve has equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (0)x + (4)y + (0) = 0&lt;/math&gt;

Curve is quadratic function:  &lt;math&gt;4y = x^2&lt;/math&gt; or &lt;math&gt;y = \frac{x^2}{4}&lt;/math&gt;

For a quick check select some random points on the curve:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
====Gallery====
{{RoundBoxTop|theme=2}}
Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = -23&lt;/math&gt;

* Focus: &lt;math&gt;(7,-21)&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0&lt;/math&gt; or &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 12&lt;/math&gt;

* Focus: &lt;math&gt;(10,-7)&lt;/math&gt;

* Equation: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0&lt;/math&gt; or &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(6.6, 6.2)&lt;/math&gt;

* Equation: &lt;math&gt;-(0.64)x^2 - (0.36)y^2 - (0.96)xy + (15.6)x + (9.2)y - (78) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0324parabola01.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;
File:0324parabola02.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;
File:0324parabola03.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;&lt;math&gt;-(0.64)x^2 - (0.36)y^2&lt;/math&gt;&lt;math&gt;- (0.96)xy + (15.6)x&lt;/math&gt;&lt;math&gt;+ (9.2)y - (78) = 0&lt;/math&gt;
&lt;/gallery&gt;
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===Ellipse===
Every ellipse has eccentricity &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;
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[[File:0325ellipse01.png|thumb|400px|'''Ellipse with ecccentricity of 0.25 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (0, 3.87298334620741688517926539978).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{4}{16} = 0.25.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 0.25.&lt;/math&gt;
]]
A simple ellipse:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = -1,0&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(1)x + (0)y + 16 = 0&lt;/math&gt; or &lt;math&gt;x = -16.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 0.25&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.9375)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (15) = 0
&lt;/syntaxhighlight&gt;
Ellipse has center at origin and equation: &lt;math&gt;(0.9375)x^2 + (1)y^2 = (15).&lt;/math&gt;


Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

points = (
    (-4  ,  0    ),
    (-3.5, -1.875),
    ( 3.5,  1.875),
    (-1  ,  3.75 ),
    ( 1  , -3.75 ),
)

A,B,F = -0.9375, -1, 15

for (x,y) in points :
    # Verify that point is on curve.
    (A*x**2 + B*y**2 + F) and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-4, 0)
    distance_to_focus, distance_to_directrix, e (3.0, 12, 0.25)
x,y (-3.5, -1.875)
    distance_to_focus, distance_to_directrix, e (3.125, 12.5, 0.25)
x,y (3.5, 1.875)
    distance_to_focus, distance_to_directrix, e (4.875, 19.5, 0.25)
x,y (-1, 3.75)
    distance_to_focus, distance_to_directrix, e (3.75, 15.0, 0.25)
x,y (1, -3.75)
    distance_to_focus, distance_to_directrix, e (4.25, 17.0, 0.25)
&lt;/syntaxhighlight&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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[[File:0325ellipse02.png|thumb|400px|'''Ellipses with ecccentricities from 0.1 to 0.9.'''
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;0,&lt;/math&gt; shape of ellipse approaches shape of circle.
&lt;/br&gt;
As eccentricity approaches &lt;math&gt;1,&lt;/math&gt; shape of ellipse approaches shape of parabola.
]]
The effect of eccentricity.


All ellipses in diagram have:

* Focus at point &lt;math&gt;(-1,0)&lt;/math&gt;

* Directrix with equation &lt;math&gt;x = -16.&lt;/math&gt;


Five ellipses are shown with eccentricities varying from  &lt;math&gt;0.1&lt;/math&gt; to &lt;math&gt;0.9.&lt;/math&gt;
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====Gallery====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;x = -10&lt;/math&gt;

* Focus: &lt;math&gt;(3,0)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.5&lt;/math&gt;

* Equation: &lt;math&gt;(-0.75)x^2 + (-1)y^2 + (0)xy + (11)x + (0)y + (16) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;y = -12&lt;/math&gt;

* Focus: &lt;math&gt;(7,-4)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.7&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (-0.51)y^2 + (0)xy + (14)x + (3.76)y + (5.56) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.6)x - (0.8)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(8,5)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 0.9&lt;/math&gt;

* Equation: &lt;math&gt;(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0325ellipse03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Ellipse on X axis.
File:0325ellipse04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Ellipse parallel to Y axis.
File:0325ellipse05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Ellipse with random orientation.
&lt;/gallery&gt;
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===Hyperbola===
Every hyperbola has eccentricity &lt;math&gt;e &gt; 1.&lt;/math&gt;
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[[File:0326hyperbola01.png|thumb|400px|'''Hyperbola with eccentricity of 1.5 and center at origin.'''
&lt;/br&gt;
Point1 &lt;math&gt;= (22.5, 21).&lt;/math&gt;&lt;/br&gt;
Eccentricity &lt;math&gt;e = \frac{\text{distance from point1 to focus}}{\text{distance from point1 to directrix}} = \frac{37.5}{25} = 1.5.&lt;/math&gt;&lt;/br&gt;
For every point on curve, &lt;math&gt;e = 1.5.&lt;/math&gt;
]]
A simple hyperbola:

Let focus be point &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;p,q = 0,-9&lt;/math&gt;

Let directrix have equation: &lt;math&gt;(0)x + (1)y + 4 = 0&lt;/math&gt; or &lt;math&gt;y = -4.&lt;/math&gt;

Let eccentricity &lt;math&gt;e = 1.5&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0
&lt;/syntaxhighlight&gt;
Hyperbola has center at origin and equation: &lt;math&gt;(1.25)y^2 - x^2 = 45.&lt;/math&gt;

Some basic checking:

&lt;syntaxhighlight lang=python&gt;
# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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&lt;math&gt;(1.25)y^2 - x^2 = 45&lt;/math&gt;

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[[File:0326hyperbola02.png|thumb|400px|'''Hyperbolas with ecccentricities from 1.5 to 20.'''
&lt;/br&gt;
As eccentricity increases, curve approaches directrix: &lt;math&gt;y = -4.&lt;/math&gt;
]]
The effect of eccentricity.


All hyperbolas in diagram have:

* Focus at point &lt;math&gt;(0,-9)&lt;/math&gt;

* Directrix with equation &lt;math&gt;y = -4.&lt;/math&gt;


Six hyperbolas are shown with eccentricities varying from  &lt;math&gt;1.5&lt;/math&gt; to &lt;math&gt;20.&lt;/math&gt;
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====Gallery====
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Curve in Figure 1 below has:

* Directrix: &lt;math&gt;y = 6&lt;/math&gt;

* Focus: &lt;math&gt;(0,1)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.5&lt;/math&gt;

* Equation: &lt;math&gt;(-1)x^2 + (1.25)y^2 + (0)xy + (0)x + (-25)y + (80) = 0&lt;/math&gt;

Curve in Figure 2 below has:

* Directrix: &lt;math&gt;x = 1&lt;/math&gt;

* Focus: &lt;math&gt;(-5,6)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 2.5&lt;/math&gt;

* Equation: &lt;math&gt;(5.25)x^2 + (-1)y^2 + (0)xy + (-22.5)x + (12)y + (-54.75) = 0&lt;/math&gt;

Curve in Figure 3 below has:

* Directrix: &lt;math&gt;(0.8)x + (0.6)y + (2.0) = 0&lt;/math&gt;

* Focus: &lt;math&gt;(-28,12)&lt;/math&gt;

* Eccentricity: &lt;math&gt;e = 1.2&lt;/math&gt;

* Equation: &lt;math&gt;(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0&lt;/math&gt;
&lt;gallery&gt;
File:0326hyperbola03.png|&lt;small&gt;Figure 1.&lt;/small&gt;&lt;/br&gt;Hyperbola on Y axis.
File:0326hyperbola04.png|&lt;small&gt;Figure 2.&lt;/small&gt;&lt;/br&gt;Hyperbola parallel to x axis.
File:0326hyperbola05.png|&lt;small&gt;Figure 3.&lt;/small&gt;&lt;/br&gt;Hyperbola with random orientation.
&lt;/gallery&gt;
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==Reversing the process==
The expression &quot;reversing the process&quot; means calculating the values of &lt;math&gt;e,&lt;/math&gt; focus and directrix when given
the equation of the conic section, the familiar values &lt;math&gt;A,B,C,D,E,F.&lt;/math&gt;

Consider the equation of a simple ellipse: &lt;math&gt;0.9375 x^2 + y^2 = 15.&lt;/math&gt;
This is a conic section where &lt;math&gt;A,B,C,D,E,F = -0.9375, -1, 0, 0, 0, 15.&lt;/math&gt;

This ellipse may be expressed as &lt;math&gt;15 x^2 + 16 y^2 = 240,&lt;/math&gt; a format more appealing to the eye
than numbers containing fractions or decimals.

However, when this ellipse is expressed as &lt;math&gt;-0.9375x^2 - y^2 + 15 = 0,&lt;/math&gt; this format is the ellipse expressed in &quot;standard form,&quot;
a notation that greatly simplifies the calculation of &lt;math&gt;a,b,c,e,p,q.&lt;/math&gt;
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Modify the equations for &lt;math&gt;A,B,C&lt;/math&gt; slightly:

&lt;math&gt;KA = Xaa - 1&lt;/math&gt; or &lt;math&gt;Xaa = KA + 1\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;KB = Xbb - 1&lt;/math&gt; or &lt;math&gt;Xbb = KB + 1\ \dots\ (2)&lt;/math&gt;

&lt;math&gt;KC = 2Xab\ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)\ \text{squared:}\ KKCC = 4XaaXbb\ \dots\ (4)&lt;/math&gt;

In &lt;math&gt;(4)&lt;/math&gt; substitute for &lt;math&gt;Xaa, Xbb:&lt;/math&gt; &lt;math&gt;C^2 K^2 = 4(KA+1)(KB+1)\ \dots\ (5)&lt;/math&gt;

&lt;math&gt;(5)&lt;/math&gt; is a quadratic equation in &lt;math&gt;K:\ (a\_)K^2 + (b\_) K + (c\_) = 0&lt;/math&gt; where:

&lt;math&gt;a\_ = 4AB - C^2&lt;/math&gt;

&lt;math&gt;b\_ = 4(A+B)&lt;/math&gt;

&lt;math&gt;c\_ = 4&lt;/math&gt;

Because &lt;math&gt;(5)&lt;/math&gt; is a quadratic equation, the solution of &lt;math&gt;(5)&lt;/math&gt; may contain a spurious value of &lt;math&gt;K&lt;/math&gt;
that will be eliminated later.

From &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2):&lt;/math&gt;

&lt;math&gt;Xaa + Xbb = KA + KB + 2&lt;/math&gt;

&lt;math&gt;X(aa + bb) = KA + KB + 2&lt;/math&gt;

Because &lt;math&gt;aa + bb = 1,\ X = KA + KB + 2&lt;/math&gt;
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==Implementation==
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&lt;syntaxhighlight lang=python&gt;
# python code


def solve_quadratic (abc) :
    '''
result = solve_quadratic (abc)
result may be :
    []
    [ root1 ]
    [ root1, root2 ]
'''
    a,b,c = abc

    if a == 0 : return [ -c/b ]
    
    disc = b**2 - 4*a*c
    
    if disc &lt; 0 : return []

    two_a = 2*a
    if disc == 0 : return [ -b/two_a ]

    root = disc.sqrt()
    r1,r2 = (-b - root)/two_a, (-b + root)/two_a
    return [r1,r2]


def calculate_Kab (ABC, flag=0) :
    '''
result = calculate_Kab (ABC)
result may be :
    []
    [tuple1]
    [tuple1,tuple2]
'''
    thisName = 'calculate_Kab (ABC, {}) :'.format(bool(flag))
    A_,B_,C_ = [ dD(str(v)) for v in ABC ]
    # Quadratic function in K: (a_)K**2 + (b_)K + (c_) = 0
    a_ = 4*A_*B_ - C_*C_
    b_ = 4*(A_+B_)
    c_ = 4

    values_of_K = solve_quadratic ((a_,b_,c_))

    if flag :
        print (thisName)
        str1 = '  A_,B_,C_' ; print (str1,eval(str1))
        str1 = '  a_,b_,c_' ; print (str1,eval(str1))
        print ('  y = ({})x^2 + ({})x + ({})'.format( float(a_), float(b_), float(c_) ))
        str1 = '  values_of_K' ; print (str1,eval(str1))

    output = []
    for K in values_of_K :
        A,B,C = [ reduce_Decimal_number(v*K) for v in (A_,B_,C_) ]
        X = A + B + 2
        if X &lt;= 0 :
            # Here is one place where the spurious value of K may be eliminated.
            if flag : print ('  K = {}, X = {}, continuing.'.format(K, X))
            continue

        aa = reduce_Decimal_number((A + 1)/X)

        if flag :
            print ('  K =', K)
            for strx in ('A', 'B', 'C', 'X', 'aa') :
                print ('   ', strx, eval(strx))

        if aa == 0 :
            a = dD(0) ; b = dD(1)
        else :
            a = aa.sqrt() ; b = C/(2*X*a)
        
        Kab = [ reduce_Decimal_number(v) for v in (K,a,b) ]
        output += [ Kab ]
    if flag:
        print (thisName)
        for t in range (0, len(output)) :
            str1 = '  output[{}] = {}'.format(t,output[t])
            print (str1)
    return output
&lt;/syntaxhighlight&gt;
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==More calculations==
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The values &lt;math&gt;D,E,F:&lt;/math&gt;

&lt;math&gt;D = 2p + 2Xac;\ 2p = (D - 2Xac)&lt;/math&gt;

&lt;math&gt;E = 2q + 2Xbc;\ 2q = (E - 2Xbc)&lt;/math&gt;

&lt;math&gt;F = Xcc - pp - qq\ \dots\ (10)&lt;/math&gt;

&lt;math&gt;(10)*4:\ 4F = 4Xcc - 4pp - 4qq\ \dots\ (11)&lt;/math&gt;

In &lt;math&gt;(11)&lt;/math&gt; replace &lt;math&gt;4pp, 4qq:\ 4F = 4Xcc - (D - 2Xac)(D - 2Xac) - (E - 2Xbc)(E - 2Xbc)\ \dots\ (12)&lt;/math&gt;

Expand &lt;math&gt;(12),&lt;/math&gt; simplify, gather like terms and result is quadratic function in &lt;math&gt;c:&lt;/math&gt;

&lt;math&gt;(a\_)c^2 + (b\_)c + (c\_) = 0\ \dots\ (14)&lt;/math&gt; where:

&lt;math&gt;a\_ = 4X(1 - Xaa - Xbb)&lt;/math&gt;

&lt;math&gt;aa + bb = 1,&lt;/math&gt; Therefore:

&lt;math&gt;a\_ = 4X(1 - X)&lt;/math&gt;

&lt;math&gt;b\_ = 4X(Da + Eb)&lt;/math&gt;

&lt;math&gt;c\_ = -(D^2 + E^2 + 4F)&lt;/math&gt;

For parabola, there is one value of &lt;math&gt;c&lt;/math&gt; because there is one directrix.

For ellipse and hyperbola, there are two values of &lt;math&gt;c&lt;/math&gt; because there are two directrices.
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===Implementation===
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&lt;syntaxhighlight lang=python&gt;
# python code

def compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :
    '''
status = compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2)

This function compares the two conic sections.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 12 = 0&quot; compare as equal.
&quot;0.75x^2 + y^2 + 3 = 0&quot; and &quot;3x^2 + 4y^2 + 10 = 0&quot; compare as not equal.

(0.24304)x^2 + (1.49296)y^2 + (-4.28544)xy + (159.3152)x + (-85.1136)y + (2858.944) = 0
            and
(-0.0784)x^2 + (-0.4816)y^2 + (1.3824)xy + (-51.392)x + (27.456)y + (-922.24) = 0
            are verified as the same curve.

&gt;&gt;&gt; abcdef1 = (0.24304, 1.49296, -4.28544, 159.3152, -85.1136, 2858.944)
&gt;&gt;&gt; abcdef2 = (-0.0784, -0.4816, 1.3824, -51.392, 27.456, -922.24)
&gt;&gt;&gt; [ (v[0]/v[1]) for v in zip(abcdef1, abcdef2) ]
[-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]
set ([-3.1, -3.1, -3.1, -3.1, -3.1, -3.1]) = {-3.1}
'''
    thisName = 'compare_ABCDEF1_ABCDEF2 (ABCDEF1, ABCDEF2) :'

    # For each value in ABCDEF1, ABCDEF2, both value1 and value2 must be 0
    # or both value1 and value2 must be non-zero.
    for v1,v2 in zip (ABCDEF1, ABCDEF2) :
        status = (bool(v1) == bool(v2))
        if not status :
            print (thisName)
            print ('  mismatch:',v1,v2)
            return status

    # Results of v1/v2 must all be the same.
    set1 = { (v1/v2) for (v1,v2) in zip (ABCDEF1, ABCDEF2) if v2 }
    status = (len(set1) == 1)
    if status : quotient, = list(set1)
    else : quotient = '??'
    
    L1 = [] ; L2 = [] ; L3 = []

    for m in range (0,6) :
        bottom = ABCDEF2[m]
        if not bottom : continue
        top = ABCDEF1[m]
        L1 += [ str(top) ] ; L3 += [ str(bottom) ]
    for m in range (0,len(L1)) :
        L2 += [ (sorted( [ len(v) for v in (L1[m], L3[m]) ] ))[-1] ] # maximum value.
    for m in range (0,len(L1)) :
        max = L2[m]
        L1[m] = ( (' '*max)+L1[m] )[-max:] # string right justified.
        L2[m] = ( '-'*max )
        L3[m] = ( (' '*max)+L3[m] )[-max:] # string right justified.
    print ('   ', '   '.join(L1))
    print ('   ', ' = '.join(L2), '=', quotient)
    print ('   ', '   '.join(L3))
    return status


def calculate_abc_epq (ABCDEF_, flag = 0) :
    '''
result = calculate_abc_epq (ABCDEF_ [, flag])
For parabola, result is:
    [((a,b,c), (e,p,q))]
For ellipse or hyperbola, result is:
    [((a1,b1,c1), (e,p1,q1)), ((a2,b2,c2), (e,p2,q2))]
'''

    thisName = 'calculate_abc_epq (ABCDEF, {}) :'.format(bool(flag))

    ABCDEF = [ dD(str(v)) for v in ABCDEF_ ]
    if flag :
        v1,v2,v3,v4,v5,v6 = ABCDEF
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(v1,v2,v3,v4,v5,v6)
        print('\n' + thisName, 'enter')
        print(str1)

    result = calculate_Kab (ABCDEF[:3], flag)

    output = []
    for (K,a,b) in result :
        A,B,C,D,E,F = [ reduce_Decimal_number(K*v) for v in ABCDEF ]

        X = A + B + 2
        e = X.sqrt()
        # Quadratic function in c: (a_)c**2 + (b_)c + (c_) = 0
        # Directrix has equation: ax + by + c = 0.
        a_ = 4*X*( 1 - X  )
        b_ = 4*X*( D*a + E*b )
        c_ = -D*D - E*E - 4*F
        
        values_of_c = solve_quadratic((a_,b_,c_))
        # values_of_c may be empty in which case this value of K is not used.

        for c in values_of_c :
            p = (D - 2*X*a*c)/2
            q = (E - 2*X*b*c)/2
            abc = [ reduce_Decimal_number(v) for v in (a,b,c) ]
            epq = [ reduce_Decimal_number(v) for v in (e,p,q) ]
            output += [ (abc,epq) ]
        if flag :
            print (thisName)
            str1 = '  ({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
            print (str1)
            if values_of_c : str1 = '  K = {}. values_of_c = {}'.format(K, values_of_c)
            else : str1 = '  K = {}. values_of_c = {}'.format(K, 'EMPTY')
            print (str1)
    if len(output) not in (1,2) :
        # This should be impossible.
        print (thisName)
        print ('  Internal error: len(output) =', len(output))
        1/0

    if flag :
        # Check output and print results.

        L1 = []
        for ((a,b,c),(e,p,q)) in output :
            print ('  e =',e)
            print ('    directrix: ({})x + ({})y + ({}) = 0'.format(a,b,c) )
            print ('    for focus : p, q = {}, {}'.format(p,q))
            # A small circle at focus for grapher.
            print ('    (x - ({}))^2 + (y - ({}))^2 = 1'.format(p,q))

            # normal through focus :
            a_,b_ = b,-a
            # normal through focus : a_ x + b_ y + c_ = 0
            c_ = reduce_Decimal_number(-(a_*p + b_*q))
            print ('    normal through focus: ({})x + ({})y + ({}) = 0'.format(a_,b_,c_) )
            L1 += [ (a_,b_,c_) ]

            _ABCDEF = ABCDEF_from_abc_epq ((a,b,c),(e,p,q))
            # This line checks that values _ABCDEF, ABCDEF make sense when compared against each other.
            if not compare_ABCDEF1_ABCDEF2 (_ABCDEF, ABCDEF) :
                print ('    _ABCDEF =',_ABCDEF)
                print ('    ABCDEF =',ABCDEF)
                2/0
        # This piece of code checks that normal through one focus is same as normal through other focus.
        # Both of these normals, if there are 2, should be same line.
        # It also checks that 2 directrices, if there are 2, are parallel.
        set2 = set(L1)
        if len(set2) != 1 :
            print ('    set2 =',set2)
            3/0

    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
==Examples==
===Parabola===

&lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0420parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y +3175 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of parabola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;16x^2 + 9y^2 - 24xy + 410x - 420y + 3175 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, focus, directrix.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 16, 9, -24, 410, -420, 3175 )

(abc,epq), = calculate_abc_epq  (input)

s1 = 'abc' ; print (s1, eval(s1))
s1 = 'epq' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc [Decimal('0.6'), Decimal('0.8'), Decimal('3')]
epq [Decimal('1'), Decimal('-10'), Decimal('6')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix: &lt;math&gt;0.6x + 0.8y + 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1&lt;/math&gt;

Focus: &lt;math&gt;p,q = -10,6&lt;/math&gt;

Because eccentricity is &lt;math&gt;1,&lt;/math&gt; curve is parabola.

Because curve is parabola, there is one directrix and one focus.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(16)x^2 + (9)y^2 + (-24)xy + (410)x + (-420)y + (3175) = 0 # This equation of parabola is not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('16'), Decimal('9'), Decimal('-24'))
  a_,b_,c_ (Decimal('0'), Decimal('100'), 4)
  y = (0.0)x^2 + (100.0)x + (4.0)
  values_of_K [Decimal('-0.04')]
  K = -0.04
    A -0.64
    B -0.36
    C 0.96
    X 1.00
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.04'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  (-0.64)x^2 + (-0.36)y^2 + (0.96)xy + (-16.4)x + (16.8)y + (-127) = 0 # This is equation of parabola in standard form.
  K = -0.04. values_of_c = [Decimal('3')]
  e = 1
    directrix: (0.6)x + (0.8)y + (3) = 0
    for focus : p, q = -10, 6
    (x - (-10))^2 + (y - (6))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (11.6) = 0
    # This is proof that equation supplied and equation in standard form are same curve.
    -0.64   -0.36   0.96   -16.4   16.8   -127
    ----- = ----- = ---- = ----- = ---- = ---- = -0.04 # K
       16       9    -24     410   -420   3175
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Ellipse===

&lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of ellipse is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;481x^2 + 369y^2 - 384xy + 5190x + 5670y + 7650 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 481, 369, -384, 5190, 5670, 7650 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('0.8'), Decimal('-3'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('37')]
epq2 [Decimal('0.8'), Decimal('-18.36'), Decimal('-23.48')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = -3, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y + 37 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 0.8&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = -18.36, -23.48&lt;/math&gt;

Because eccentricity is &lt;math&gt;0.8,&lt;/math&gt; curve is ellipse.

Because curve is ellipse, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
(481)x^2 + (369)y^2 + (-384)xy + (5190)x + (5670)y + (7650) = 0 # Not in standard form.
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('481'), Decimal('369'), Decimal('-384'))
  a_,b_,c_ (Decimal('562500'), Decimal('3400'), 4)
  y = (562500.0)x^2 + (3400.0)x + (4.0)
  values_of_K [Decimal('-0.004444444444444444444444'), Decimal('-0.0016')]
  # Unwanted value of K is rejected here.
  K = -0.004444444444444444444444, X = -1.777777777777777777778, continuing.
  K = -0.0016
    A -0.7696
    B -0.5904
    C 0.6144
    X 0.6400
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('-0.0016'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Equation of ellipse in standard form.
  (-0.7696)x^2 + (-0.5904)y^2 + (0.6144)xy + (-8.304)x + (-9.072)y + (-12.24) = 0
  K = -0.0016. values_of_c = [Decimal('-3'), Decimal('37')]
  e = 0.8
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = -3, -3
    (x - (-3))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
  e = 0.8
    directrix: (0.6)x + (0.8)y + (37) = 0
    for focus : p, q = -18.36, -23.48
    (x - (-18.36))^2 + (y - (-23.48))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (0.6) = 0 # Same as normal above.
    # Method calculates equation of ellipse using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and supplied values are the same curve.
    -0.7696   -0.5904   0.6144   -8.304   -9.072   -12.24
    ------- = ------- = ------ = ------ = ------ = ------ = -0.0016 # K
        481       369     -384     5190     5670     7650
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Hyperbola===

&lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
Given equation of conic section: &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;

Calculate &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

input = ( 7, 0, -24, 90, 216, -81 )

(abc1,epq1),(abc2,epq2) = calculate_abc_epq  (input)

s1 = 'abc1' ; print (s1, eval(s1))
s1 = 'epq1' ; print (s1, eval(s1))
s1 = 'abc2' ; print (s1, eval(s1))
s1 = 'epq2' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
abc1 [Decimal('0.6'), Decimal('0.8'), Decimal('-3')]
epq1 [Decimal('1.25'), Decimal('0'), Decimal('-3')]
abc2 [Decimal('0.6'), Decimal('0.8'), Decimal('-22.2')]
epq2 [Decimal('1.25'), Decimal('18'), Decimal('21')]
&lt;/syntaxhighlight&gt;
interpreted as:

Directrix 1: &lt;math&gt;0.6x + 0.8y - 3 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 1: &lt;math&gt;p,q = 0, -3&lt;/math&gt;

Directrix 2: &lt;math&gt;0.6x + 0.8y - 22.2 = 0&lt;/math&gt;

Eccentricity: &lt;math&gt;e = 1.25&lt;/math&gt;

Focus 2: &lt;math&gt;p,q = 18, 21&lt;/math&gt;

Because eccentricity is &lt;math&gt;1.25,&lt;/math&gt; curve is hyperbola.

Because curve is hyperbola, there are two directrices and two foci.

For more insight into the method of calculation and also to check the calculation:
&lt;syntaxhighlight lang=python&gt;
calculate_abc_epq  (input, 1) # Set flag to 1.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
calculate_abc_epq (ABCDEF, True) : enter
# Given equation is not in standard form.
(7)x^2 + (0)y^2 + (-24)xy + (90)x + (216)y + (-81) = 0
calculate_Kab (ABC, True) :
  A_,B_,C_ (Decimal('7'), Decimal('0'), Decimal('-24'))
  a_,b_,c_ (Decimal('-576'), Decimal('28'), 4)
  y = (-576.0)x^2 + (28.0)x + (4.0)
  values_of_K [Decimal('0.1111111111111111111111'), Decimal('-0.0625')]
  K = 0.1111111111111111111111
    A 0.7777777777777777777777
    B 0
    C -2.666666666666666666666
    X 2.777777777777777777778
    aa 0.64
  K = -0.0625
    A -0.4375
    B 0
    C 1.5
    X 1.5625
    aa 0.36
calculate_Kab (ABC, True) :
  output[0] = [Decimal('0.1111111111111111111111'), Decimal('0.8'), Decimal('-0.6')]
  output[1] = [Decimal('-0.0625'), Decimal('0.6'), Decimal('0.8')]
calculate_abc_epq (ABCDEF, True) :
  # Here is where unwanted value of K is rejected.
  (0.7777777777777777777777)x^2 + (0)y^2 + (-2.666666666666666666666)xy + (10)x + (24)y + (-9) = 0
  K = 0.1111111111111111111111. values_of_c = EMPTY
calculate_abc_epq (ABCDEF, True) :
  # Equation of hyperbola in standard form.
  (-0.4375)x^2 + (0)y^2 + (1.5)xy + (-5.625)x + (-13.5)y + (5.0625) = 0
  K = -0.0625. values_of_c = [Decimal('-3'), Decimal('-22.2')]
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-3) = 0
    for focus : p, q = 0, -3
    (x - (0))^2 + (y - (-3))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
  e = 1.25
    directrix: (0.6)x + (0.8)y + (-22.2) = 0
    for focus : p, q = 18, 21
    (x - (18))^2 + (y - (21))^2 = 1
    normal through focus: (0.8)x + (-0.6)y + (-1.8) = 0 # Same as normal above.
    # Method calculates equation of hyperbola using these values of directrix, eccentricity and focus.
    # Method then verifies that calculated and given values are the same curve.
    -0.4375   1.5   -5.625   -13.5   5.0625
    ------- = --- = ------ = ----- = ------ = -0.0625 # K
          7   -24       90     216      -81
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


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[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Parabola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Ellipse====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
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[[File:0421hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;7x^2 + 0y^2 - 24xy + 90x + 216y - 81 = 0.&lt;/math&gt;'''
&lt;/br&gt;
Equation of hyperbola is given.&lt;/br&gt;
This section calculates &lt;math&gt;\text{eccentricity, foci, directrices.}&lt;/math&gt;
]]
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=Two Conic Sections=

Examples of conic sections include: ellipse, circle, parabola and hyperbola.

This section presents examples of two conic sections, circle and ellipse, and how to calculate 
the coordinates of the point/s of intersection, if any, of the two sections.

Let one section with name &lt;math&gt;ABCDEF&lt;/math&gt; have equation
&lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.&lt;/math&gt;

Let other section with name &lt;math&gt;abcdef&lt;/math&gt; have equation
&lt;math&gt;ax^2 + by^2 + cxy + dx + ey + f = 0.&lt;/math&gt;

Because there can be as many as 4 points of intersection, a special &quot;resolvent&quot; quartic function
is used to calculate the &lt;math&gt;x&lt;/math&gt; coordinates of the point/s of intersection.

Coefficients of associated &quot;resolvent&quot; quartic are calculated as follows:

&lt;syntaxhighlight lang=python&gt;
# python code

def intersection_of_2_conic_sections (abcdef, ABCDEF) :
    '''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
    '''
    A,B,C,D,E,F = ABCDEF
    a,b,c,d,e,f = abcdef
    
    G = ((-1)*(B)*(a) + (1)*(A)*(b))
    H = ((-1)*(B)*(d) + (1)*(D)*(b))
    I = ((-1)*(B)*(f) + (1)*(F)*(b))
    J = ((-1)*(C)*(a) + (1)*(A)*(c))
    K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
    L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
    M = ((-1)*(E)*(f) + (1)*(F)*(e))
    g = ((-1)*(C)*(b) + (1)*(B)*(c))
    h = ((-1)*(E)*(b) + (1)*(B)*(e))
    i = ((-1)*(A)*(b) + (1)*(B)*(a))
    j = ((-1)*(D)*(b) + (1)*(B)*(d))
    k = ((-1)*(F)*(b) + (1)*(B)*(f))

    A_ =   ((-1)*(J)*(g) + (1)*(G)*(i))
    B_ =   ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
    C_ =   ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
    D_ =   ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
    E_ =   ((-1)*(M)*(h) + (1)*(I)*(k))

    str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
    print (str1)
    
    return A_,B_,C_,D_,E_
&lt;/syntaxhighlight&gt;

&lt;math&gt;y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876&lt;/math&gt;


In cartesian coordinate geometry of three dimensions a sphere is represented by the equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[ (13,  7, 20,  1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=quartic=
A close examination of coefficients &lt;math&gt;R, S&lt;/math&gt; shows that both coefficients are always
exactly divisible by &lt;math&gt;4.&lt;/math&gt;

Therefore, all coefficients may be defined as follows:

&lt;math&gt;P = 1&lt;/math&gt;

&lt;math&gt;Q = A2&lt;/math&gt;

&lt;math&gt;R = \frac{A2^2 - C}{4}&lt;/math&gt;

&lt;math&gt;S = \frac{-B4^2}{4}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


The value &lt;math&gt;Rs - Sr&lt;/math&gt; is in fact:

&lt;syntaxhighlight&gt;
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee 
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd 
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee 
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde 
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee 
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde 
- 3aabbccdddddd
&lt;/syntaxhighlight&gt;
which, by removing values &lt;math&gt;aa, ad&lt;/math&gt; (common to all values), may be reduced to:
&lt;syntaxhighlight&gt;
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee 
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd 
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee 
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde 
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee 
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde 
- 3bbccdddd
)
&lt;/syntaxhighlight&gt;
If &lt;math&gt;status == 0,&lt;/math&gt; there are at least 2 equal roots which may be calculated as shown below.
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If coefficient &lt;math&gt;d&lt;/math&gt; is non-zero, it is not necessary to calculate &lt;math&gt;status.&lt;/math&gt;

If coefficient &lt;math&gt;d == 0,&lt;/math&gt; verify that &lt;math&gt;status = 0&lt;/math&gt; before proceeding.
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===Examples===
&lt;math&gt;y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735&lt;/math&gt; &lt;code&gt;(quartic function)&lt;/code&gt;

&lt;math&gt;y' = g(x) = 4x^3 + 18x^2 - 96x - 182&lt;/math&gt; &lt;code&gt;(cubic function (2a), derivative)&lt;/code&gt;

&lt;math&gt;y        =        -182x^3        -        4032x^2      -     4494x      +      103684&lt;/math&gt; &lt;code&gt;(cubic function (1a))&lt;/code&gt;

&lt;math&gt;y = -12852x^2 - 35448x + 381612&lt;/math&gt; &lt;code&gt;(quadratic function (1b))&lt;/code&gt;

&lt;math&gt;y  =  -381612x^2  -  1132488x  +  10771572&lt;/math&gt; &lt;code&gt;(quadratic function (2b))&lt;/code&gt;

&lt;math&gt;y = 7191475200x + 50340326400&lt;/math&gt; &lt;code&gt;(linear function (2c))&lt;/code&gt;

&lt;math&gt;y = -1027353600x - 7191475200&lt;/math&gt; &lt;code&gt;(linear function (1c))&lt;/code&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

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Python function &lt;code&gt;equalRoots()&lt;/code&gt; below implements &lt;code&gt;status&lt;/code&gt; as presented under 
[https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above.
&lt;syntaxhighlight lang=python&gt;
# python code

def equalRoots(abcde) :
    '''
This function returns True if quartic function contains at least 2 equal roots.
    '''
    a,b,c,d,e = abcde
    
    aa = a*a ; aaa = aa*a
    bb = b*b ; bbb = bb*b ; bbbb = bb*bb
    cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
    dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
    ee = e*e ; eee = ee*e ; eeee = ee*ee
    
    v1 = (
            +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
            +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
            +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
            +20*bb*ccc*dd*e
         )
    v2 = (
            -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
            -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
            -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
         )
    return (v1+v2) == 0

t1 = (
    ((1, -1, -19, -11,   30),  '4 unique, real roots.'),
    ((4,  4,-119, -60,  675),  '4 unique, real roots, B4 = 0.'),
    ((1,  6, -48,-182,  735),  '2 equal roots.'),
    ((1,-12,  50, -84,   45),  '2 equal roots. B4 = 0.'),
    ((1,-20, 146,-476,  637),  '2 equal roots, 2 complex roots.'),
    ((1,-12,  58,-132,  117),  '2 equal roots, 2 complex roots.  B4 = 0.'),
    ((1, -2, -36, 162, -189),  '3 equal roots.'),
    ((1,-20, 150,-500,  625),  '4 equal roots.'),
    ((1, -6, -11,  60,  100),  '2 pairs of equal roots, B4 = 0.'),
    ((4,  4, -75,-776,-1869),  '2 complex roots.'),
    ((1,-12,  33,  18, -208),  '2 complex roots, B4 = 0.'), 
    ((1,-20, 408,2296,18020),  '4 complex roots.'),
    ((1,-12,  83, -282, 442),  '4 complex roots, B4 = 0.'),
    ((1,-12,  62,-156,  169),  '2 pairs of equal complex roots, B4 = 0.'),
)

for v in t1 :
    abcde, comment = v
    print ()
    fourRoots = rootsOfQuartic (abcde)
    print (comment)
    print ('    Coefficients =', abcde)
    print ('    Four roots =', fourRoots)
    print ('    Equal roots detected:', equalRoots(abcde))
    # Check results.
    a,b,c,d,e = abcde
    for x in fourRoots :
        # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
        # This test tolerates small rounding errors sometimes caused
        # by the limited precision of python floating point numbers.
        sum = a*x**4 + b*x**3 + c*x**2 + d*x
        if not almostEqual (sum, -e) : 1/0 # Create exception.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
4 unique, real roots.
    Coefficients = (1, -1, -19, -11, 30)
    Four roots = [5.0, 1.0, -2.0, -3.0]
    Equal roots detected: False

4 unique, real roots, B4 = 0.
    Coefficients = (4, 4, -119, -60, 675)
    Four roots = [2.5, -3.0, 4.5, -5.0]
    Equal roots detected: False

2 equal roots.
    Coefficients = (1, 6, -48, -182, 735)
    Four roots = [5.0, 3.0, -7.0, -7.0]
    Equal roots detected: True

2 equal roots. B4 = 0.
    Coefficients = (1, -12, 50, -84, 45)
    Four roots = [3.0, 3.0, 5.0, 1.0]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.
    Coefficients = (1, -20, 146, -476, 637)
    Four roots = [7.0, 7.0, (3+2j), (3-2j)]
    Equal roots detected: True
    
2 equal roots, 2 complex roots.  B4 = 0.
    Coefficients = (1, -12, 58, -132, 117)
    Four roots = [(3+2j), (3-2j), 3.0, 3.0]
    Equal roots detected: True

3 equal roots.
    Coefficients = (1, -2, -36, 162, -189)
    Four roots = [3.0, 3.0, 3.0, -7.0]
    Equal roots detected: True

4 equal roots.
    Coefficients = (1, -20, 150, -500, 625)
    Four roots = [5.0, 5.0, 5.0, 5.0]
    Equal roots detected: True

2 pairs of equal roots, B4 = 0.
    Coefficients = (1, -6, -11, 60, 100)
    Four roots = [5.0, -2.0, 5.0, -2.0]
    Equal roots detected: True

2 complex roots.
    Coefficients = (4, 4, -75, -776, -1869)
    Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
    Equal roots detected: False

2 complex roots, B4 = 0.
    Coefficients = (1, -12, 33, 18, -208)
    Four roots = [(3+2j), (3-2j), 8.0, -2.0]
    Equal roots detected: False

4 complex roots.
    Coefficients = (1, -20, 408, 2296, 18020)
    Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
    Equal roots detected: False

4 complex roots, B4 = 0.
    Coefficients = (1, -12, 83, -282, 442)
    Four roots = [(3+5j), (3-5j), (3+2j), (3-2j)]
    Equal roots detected: False

2 pairs of equal complex roots, B4 = 0.
    Coefficients = (1, -12, 62, -156, 169)
    Four roots = [(3+2j), (3-2j), (3+2j), (3-2j)]
    Equal roots detected: True
&lt;/syntaxhighlight&gt;
When description contains note &lt;math&gt;B4 = 0,&lt;/math&gt; depressed quartic was processed as quadratic in &lt;math&gt;t^2.&lt;/math&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Two real and two complex roots==

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;

==gallery==


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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
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C
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;

=allEqual=

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;8&lt;/code&gt; || &lt;code&gt;-193&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;9&lt;/code&gt; || &lt;code&gt;-176&lt;/code&gt;
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;14&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-61&lt;/code&gt;
|-
| &lt;code&gt;15&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-32&lt;/code&gt;
|-
| &lt;code&gt;16&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;-1&lt;/code&gt;
|-
| &lt;code&gt;17&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;32&lt;/code&gt;
|-
| &lt;code&gt;18&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;&lt;/code&gt;&lt;code&gt;67&lt;/code&gt;
|-
| &lt;code&gt;19&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;104&lt;/code&gt;
|-
| &lt;code&gt;20&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;143&lt;/code&gt;
|-
| &lt;code&gt;21&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;184&lt;/code&gt;
|-
| &lt;code&gt;22&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;227&lt;/code&gt;
|-
| &lt;code&gt;23&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;272&lt;/code&gt;
|-
| &lt;code&gt;24&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;319&lt;/code&gt;
|-
| &lt;code&gt;25&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;368&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
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{{Information Q|Q43997847}}

== Licensing ==
{{cc-by-3.0}}
[[Category:WikiJournal]]</text>
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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      <text bytes="100" xml:space="preserve">== Summary ==
{{Information Q|Q28867140}}

== Licensing ==
{{cc-by-sa-3.0}}
[[Category:WikiJournal]]</text>
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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{{Information
|Description=WikiJournal of Medicine articles and citations 2014-2016
|Source=User generated
|Date=9 February 2017
|Author=Ear-phone
|Permission= Creative Commons Attribution ShareAlike 3.0 Unported License.
}}

== Licensing ==
{{cc-by-sa-3.0}}
[[Category:WikiJournal]]</text>
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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== Licensing ==
{{cc-by-sa-3.0}}
[[Category:WikiJournal]]</text>
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{{Information
|Description=First peer review of [[Draft:WikiJournal of Medicine/Rotavirus]]. No conflict of interests declared.
|Source=Own work by reveiwer.
|Date=2017-04-14
|Author= Anonymous. Uploaded by [[User:Mikael Häggström|Mikael Häggström]]
|Permission=
}}

== Licensing ==
{{cc-by-3.0}}
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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{{Information
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|Source=Authors
|Date=January 20th, 2017
|Author=S. Bhaumik, Z. Lassi (authors) - M. Laurent (Editor)
|Permission=authors and Editor have given permission for cc-by-sa-3.0
}}
== Licensing ==
{{cc-by-sa-3.0}}
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{{Information
|Description=WikiJournal Author reply to Initial Editor feedback to meta-analysis on vitamin D for adjunct CAP treatment
|Source=Authors
|Date=February 28th, 2017
|Author=S. Bhaumik, Z. Lassi (authors) - M. Laurent (Editor)
|Permission=authors and Editor have given permission for cc-by-sa-3.0
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== Licensing ==
{{cc-by-sa-3.0}}
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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{{Information
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|Source=Authors
|Date=May 1st, 2017
|Author=[[User:Stevenfruitsmaak|M. Laurent (Editor)]]
|Permission=Anonymous reviewer has given permission for cc-by-sa-3.0
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== Licensing ==
{{cc-by-3.0}}
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|Source=[[User:Stevenfruitsmaak|Steven Fruitsmaak]] &lt;small&gt;([[User_talk:Stevenfruitsmaak|Reply]])&lt;/small&gt; 21:51, 24 June 2017 (UTC)
|Date=15-6-2017
|Author=[[User:Stevenfruitsmaak|Steven Fruitsmaak]] &lt;small&gt;([[User_talk:Stevenfruitsmaak|Reply]])&lt;/small&gt; 21:51, 24 June 2017 (UTC)
|Permission=own work
}}
== Licensing ==
{{self|GFDL|cc-by-3.0}}
[[Category:WikiJournal]]</text>
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|Source = {{Cite journal|last=Bhaumik|first=Soumyadeep|last2=Lassi|first2=Zohra|title=Vitamin D as an adjunct for acute community-acquired pneumonia among infants and children: systematic review and meta-analysis|url=https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Vitamin_D_as_an_adjunct_for_acute_community-acquired_pneumonia_among_infants_and_children:_systematic_review_and_meta-analysis|journal=WikiJournal of Medicine|language=en|volume=4|issue=1|doi=10.15347/wjm/2017.005}}
|Date = 25/06/2017
|Author = Soumyadeep Bhaumik and Zohra Lassi
|Permission = See below
}}

== Licensing ==
{{Cc-by-3.0}}
[[Category:WikiJournal]]</text>
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|Description = PDF version of [[WikiJournal of Medicine/Acute gastrointestinal bleeding from a chronic cause: a teaching case report]]
|Source = {{Cite journal|last=Laurent|first=M|last2=Van Overbeke|first2=L|title=Acute gastrointestinal bleeding from a chronic cause: a teaching case report|url=https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Acute_gastrointestinal_bleeding_from_a_chronic_cause:_a_teaching_case_report|journal=WikiJournal of Medicine|date=2017|volume=4|issue=1|doi=10.15347/wjm/2017.006}}

|Date = 1 August 2017
|Author =  Michaël R. Laurent and Lode Van Overbeke
|Permission = See below
}}

== Licensing ==
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    <title>File:Rehumanise.pdf</title>
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      <text bytes="2515" xml:space="preserve">== Summary ==
{{Information
|Description=The organisers of eLearning Korea 2017 have invited me to give a talk on the future of education and educational technology. The conference has a curious byline “a happy encounter with new technology”, and it's to this byline I target the presentation.
I aim to acknowledge the unhappiness created by technology and propose humanism to ward off technocratic tyranny and to discover what technological happiness might be.
I hope my proposition is clear - that for there to be a happy encounter with technology, we need to re-orientate ourselves to humanist perspectives. Those perspectives can be found in history, philosophy, ethics, anthropology, theory, art, storytelling, questioning, criticism and debate. Sensitivity to humanism needs to be nurtured, the ember that might make a flame seems at risk of being extinguished.
It is with humility and hope that I offer this idea to the eLearning Korea 2017 Conference.
Please play this video as background: The Mother of All Demos Douglas Engelbart Et al 1968

Images used: 
# https://commons.wikimedia.org/wiki/File:After_Rain_at_Mt._Inwang.jpg
# https://commons.wikimedia.org/wiki/File:%27Scholarly_Gathering%27,_Joseon_dynasty,_c._1586,_Honolulu_Museum_of_Art_2015-36-01.jpg
# https://commons.wikimedia.org/wiki/File:Hyewon-Wolha-jeongin-2.jpg
# https://en.wikipedia.org/wiki/File:Koreans_oldest_pic_3.jpg
# TBA
# https://en.wikipedia.org/wiki/File:Lopez_scaling_seawall.jpg
# https://commons.wikimedia.org/wiki/File:1951.7_...._(7445954378).jpg
# https://en.wikipedia.org/wiki/File:Korea-Sinchuk_Jinyeonuigwe-03.jpg
# https://commons.wikimedia.org/wiki/File:Korean.painting-Miindo-1825.jpg
# https://en.wikipedia.org/wiki/File:Korea-Gyeongju-Bulguksa-33.jpg
# https://commons.wikimedia.org/wiki/File:Gyeonggi-gamyeong-do.jpg
# https://commons.wikimedia.org/wiki/File:Kim_Hong-do,_Yeombulseoseung.jpg
# https://commons.wikimedia.org/wiki/File:Jeong_Seon-Bakyeon_pokpo.jpg
# https://commons.wikimedia.org/wiki/File:HMJE_1-1.svg
# https://commons.wikimedia.org/wiki/File:Owon-Songha.noseungdo.jpg
# https://commons.wikimedia.org/wiki/File:Gyeomjae-Yangcheon-Gaehwasa.jpg

Please play this video as background: [https://youtu.be/yJDv-zdhzMY?t=1m38s The Mother of All Demos] Douglas Engelbart Et al 1968
|Source=Leigh Blackall (featuring works sourced from Wikimedia Commons attributed by links)
|Date=August 2017
|Author=Leigh Blackall
|Permission={{cc-by-4.0}}
}}

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      <text bytes="535" xml:space="preserve">== Summary ==
{{Information
|Description = PDF version of [[WikiJournal of Medicine/Rotavirus]]
|Source = {{Cite_journal|last=Beards|first=Graham|last2=et al|title=Rotavirus|url=https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Rotavirus|journal=WikiJournal of Medicine|volume=4|issue=1|doi=10.15347/wjm/2017.007}}
|Date = 10 November 2017
|Author =  Graham Beards ''[https://xtools.wmflabs.org/articleinfo/en.wikipedia.org/Rotavirus et al.]''
|Permission = See below
}}

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      <text bytes="97" xml:space="preserve">== Summary ==
{{Information Q|Q55887092}}

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    <title>File:WikiJournal of Science.Review.Jan12.2018 LEAD env.pdf</title>
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      <text bytes="305" xml:space="preserve">== Summary ==
{{Information
|Description=Peer review of final two sections of article &quot;Lead&quot; by Robert M. Gogal Jr.
|Source= provided by Robert M. Gogal Jr.
|Date=12 January 2018
|Author=Robert M. Gogal Jr.
|Permission={{cc-by-sa-3.0,2.5,2.0,1.0}}
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  <page>
    <title>Animal Farm</title>
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      <text bytes="24385" xml:space="preserve">==Characters==
*'''Mr. Jones''' - Czar Nicholas II
*'''Old Major''' - Karl Max
*'''Snowball''' - Trotsky
*'''Napoleon''' - Stalin
*'''Squealer''' - Stalin's propoganda committee
*'''Dogs''' - KGB
*'''Boxer''' - dedicated but tricked communist supporters
*'''Moses''' - religions
*'''Frederick''' - Hitler/Deutschland
*'''Pilkington''' - FDR/Churchill/Allie Forces
*'''Whymper''' - Capitalist Westerners 

==Events/Objects==
*'''Windmill''' - 5 year plans
*'''Major's Skull''' - Lenin's body
*'''Battle of the Cowshed''' - Russian Revolution
*'''Animal Executions''' - Stalin's purge
*'''Battle of the Windmill''' - WWII

==Chapter 1==
#'''For what purpose did Old Major call the meeting of the animals?'''
#*To share a dream he had.
#'''According to Major, what is the cause of all the animals' problems? Be specific.'''
#*Mankind; Man consumes without producing and only gives the bare minimum to prevent starving. He kills animals who are old and useless, they are selfish also. Sells the young offspring for profit. Humans don't know how to best use the land.
#'''After they vote and decide rats are comrades, Major summarizes his points for the animals to remember. What are they?'''
#*2 legs - enemy 
#*4 legs or wings - friend
#*Don't resemble a man, like living in a house, sleeping on a bed, touch money, etc. (no alcohol, trade, no clothes, smoke). 
#*Treat each other equally and with no tyranny.

==Chapter 2==
#'''Who gained leadership of the animals? Why?'''
#*Pigs; Most clever animals on the farm.
#'''What concept did the three pigs come up with?'''
#*Animalism
#'''How did the other animals respond to this concept? What are some of the questions the animals had for the pigs?'''
#*The other animals did not like it or understand because they did not think it would happen in their lifetimes. They want to know who will feed us after Mr. Jones is gone and why should we care about what happens after we die. Why should we work for it if it will happen anyways?
#'''Who is Moses? What idea does he spread around to the animals?'''
#*Tame Raven - Mr. Jones's pet; Religion (sugar candy mountain)
#'''What impact do Clover and Boxer have on others?'''
#*Persuaded other animals to believe in Animalism through simple arguments. 
#'''Who gained leadership of the animals? Why?'''
#*Pigs; Most cleverest animals on the farm.
#'''What concept did the three pigs come up with?'''
#*Animalism
#'''How did the other animals respond to this concept? What are some of the questions the animals had for the pigs?'''
#*The other animals did not like it or understand because they did not think it would happen in their lifetimes. They want to know who will feed us after Mr. Jones is gone and why should we care about what happens after we die. Why should we work for it if it will happen anyway?
#'''Who is Moses? What idea does he spread around to the animals?'''
#*Tame Raven--Mr. Jones's pet; religion (sugar candy mountain)
#'''What impact do Clover and Boxer have on others?'''
#*Persuaded other animals to believe in Animalism through simple arguments.
#'''What events lead up to the rebellion?'''
#*Mr. Jones started to neglect the farm, not feeding the animals, men were idle and dishonest, fields full with weed, buildings needed roofing, hedges were neglected.
#'''What do the animals do that causes the rebellion to happen?'''
#*One of the cows broke into the store and the animals began to help themselves to the bins, waking Mr. Jones up. 
#'''Describe the events that happen in the rebellion from beginning to end.'''
#*Mr. Jones wakes up. He and his men begin to lash their whips frantically.
#*The animals retaliate against the whips by jumping on them. They start to butt and kick Mr. Jones and his men. This causes Mr. Jones and men to leave the farm on foot 2 minutes later.
#*Mrs. Jones is chased by Moses.
#*Mr. Jones and his men are defeated and the Manor Farm is theirs.
#'''What are the first four things that the animals do once they win the rebellion?'''
#*Checked to make sure no human remains on the farm.
#*Throw out remains that was used/a sign of Mr. Jones's terror, like knives.
#*Napoleon leads the animals back to the store and gives them corn and 2 biscuits for the dogs.
#*They sing Beasts of England and then settled down and slept for the night.
#'''Where is Mollie discovered to be when the animals go through the house?'''
#*Mollie was behind in the best bedroom admiring herself.
#'''What have the pigs taught themselves to do? To what do they change the name of the farm to?'''
#*The pigs have taught themselves to read and write from a spelling book from Mr. Jones; The name was changed to Animal Farm.
#'''List the 7 commandments of Animalism'''
#*Whatever goes upon two legs is an enemy.
#*Whatever goes upon four legs or has wings is a friend.
#*No animal shall wear clothes.
#*No animal shall sleep in a bed.
#*No animal shall drink alcohol.
#*No animal shall kill any other animal.
#*All animals are equal.
#'''What happens to the milk?'''
#*The milk disappears.

==Chapter 3==
#'''What is different about the harvest this year? How does this represent the true idea of Communism?'''
#*The harvest was a lot more successful than with Mr. Jones and his men; Production of the farm truly belongs to the animals.
#'''Describe Boxer at the beginning of the chapter. What is his motto?'''
#*Boxer worked even harder than he worked when he was working for Mr. Jones. He is the admiration for everybody; &quot;I will work harder&quot;.
#'''How have the animals changed at the start of the chapter with their newly found freedom? Who hasn't changed?'''
#*The animals now truly love eating their food knowing that they produced it themselves without their cruel masters giving them the bare minimum. The animals work together as a team and barely argue; Mollie hasn't changed because she only cares about herself.
#'''What routine do the animal engage in on Sunday mornings? Explain the flag and its elements.'''
#*They don't do any work on Sunday and breakfast was an hour later than usual. A ceremony takes place after breakfast; The flag was an old green tablecloth painted with a hoof and horn. The hoof and horn signified the Republic of the Animals. 
#'''What is the degree of success achieved by other animals on the farm when it comes to reading?'''
#*Almost every animal on the farm was literate in some degrees by the autumn as reading an writing classes which were held at the farm were a success; Mollie only knows the letters of her name, Boxer couldn't get past the letter &quot;D&quot;, pigs knew the alphabets, dogs were behind the pigs. 
#'''What maxim was created for the farm? Why?'''
#*The maxim that was created was &quot;Four legs good, two legs bad&quot; and it was created because some of the animals (sheep, hens, ducks) could not learn the 7 commandments.
#'''Where does Napoleon take Jessie and Bluebell's pups? What do you think he is doing with them?'''
#*Napoleon took them to a loft which could only be reached by a ladder from the Larness-room; teaching them by himself.
#'''How does Squealer explain the disappearance of the milk and the apples?'''
#*He explains that the pigs need the milk and apples more as they need the brian-helping nutrients from the milk and apples more than the other animals as they're the ones who rule the other animals.
#'''What is the main argument Squealer uses to ensure what he says is taken seriously by the other animals?'''
#*The main argument: &quot;If the pigs fails, then Mr. Jones will be back!&quot;

==Chapter 4==
#'''Contrast Mr. Pilkington to Mr. Frederick'''.
#*'''Mr. Pilkington''': Represents Allied Forces, easy-going, hunted or fish by the seas, owns an unorganized farm: Foxwood.
#*'''Mr. Frederick''': Represents Germany, tough-shrewd, involved in many lawsuits, hard bargains, owns an organized farm: Pinchfield. 
#'''How do the humans respond to their own animals singing &quot;Beasts of England&quot;?'''
#*The animals are whipped on the spot if they are heard singing it.
#'''Why does no one go to help Mr. Jones reclaim his farm at the beginning of the chapter?'''
#*Everyone is secretly trying to figure out a way to benefit from Mr. Jones' bad situation.
#'''Who led the first counter-attack when Mr. Jones returned to the farm with other men to try to reclaim it? What influenced him to lead the counter-attack?'''
#*Snowball leads the counter-attack after he reads a book on Julius Cesear's military experiences. 
#'''How do the animals trick Mr. Jones and the other men?'''
#*The animals run back into the farmyard acting as if they are retreating, but they are leading the men into an ambush.
#'''What happens to Snowball? The sheep? Mollie?'''
#*Mr. Jones fires his gun at Snowball, but the pellets graze his back and kill the sheep behind him. Mollie is hiding in her stall instead of fighting.
#'''How does Boxer react to the idea of killing a human?'''
#*Boxer is saddened at the thought of killing someone, even a human. 
#'''What awards do Snowball and Boxer receive? The sheep?'''
#*Snowball + Boxer = &quot;Animal Hero, First Class&quot;
#*Dead sheep = &quot;Animal Hero, Second Class&quot;
#'''What do they do with Mr. Jones's gun?'''
#*The gun is placed at the bottom of the flagpole. It will be fired twice a year once on the anniversary of Battle of Cowshed on October 12 and Midsummer's Day for the rebellion.

==Chapter 5==
#'''What has Clover discovered about Mollie? What has she found in Mollie's stall?'''
#*Clover has found out that Mollie is interacting with the humans and Clover finds her 2 favorite things: Lump of sugar and ribbons. 
#'''What happened to Mollie? How is that treated by other animals?'''
#*Mollie runs away to joy to another farm across town so she can have sugar, wear ribbons and be a pet; They animals never speak fo Mollie against because she betrayed them. 
#'''What idea does Snowball come up with for the farm? What does he hope that it will provide for the farm? How will it make their lives better?'''
#*Windmill; Electricity created will power machines as well as produce light and heat for each stall; They will benefit from more comfort and less work.
#'''What is Napoleon's opinion of Snowball's plan? What does he do to show it?'''
#*Napoleon is not impressed by his plans; Urinates on it.
#'''During a Sunday meeting, the animals are trying to decide whether or not to go ahead and build the windmill. What is Snowball's argument? Napoleon's?'''
#*Snowball claims it will lower their work week to 3 days a week.
#*Napoleon believes its nonsense.
#'''What happens when the animals decide to go with Snowball's plan?'''
#*Napoleon has the dogs, KGB, run Snowball off the farm.
#'''What is the first decision that Napoleon makes now that he is in charge?'''
#*He abolishes the Sunday morning meetings.
#'''How will decisions be made on the farm?'''
#*By a special committee of pigs presided over by Napoleon himself.
#'''What new step has been added to the Sunday morning processional?'''
#*Animals must walk past the skull of Old Major that has been dug up and placed at the bottom of the flagpole next to Mr. Jones's gun.
#'''How has the seating arrangement changed during the meetings?'''
#*Napoleon, Squealer, and a pig named Minimus sat together on the raised platform like Old Major used to do. The dogs sit in front of the three of them and the rest of the pigs are on the back end of the platform. The separation of the pigs and dogs is indicating that their perceived superiority to the other animals.
#'''How does Napoleon use the sheep to help quiet arguments that oppose his plans?'''
#*The sheep yell repeatedly, &quot;4 legs good, 2 legs bad&quot;, over-top of anyone trying to rebel.
#'''What does Napoleon decide to do after all? How does Squealer convince the animals that Napoleon supported the plan all along?'''
#*Napoleon decides to build the windmill; He tells that it was Napoleon's idea all along and the growling of the dogs keeps the animals from denying the idea.

==Chapter 6==
#'''Reread the opening sentence of this chapter. What do you notice is significant in the diction?'''
#*The animals are working like slaves... this is the same way they were treated with Mr. Jones.
#'''What changes does Napoleon make to the work week?'''
#*They work 60 hours a week (10 hours a day) - now work on Sundays (they say it is voluntary, but if an animal does not work on Sundays, his/her food will be cut in half).
#'''What items become scarce that the farm cannot produce? How does Napoleon decide they will overcome these hardships?'''
#*Nails, iron for horseshoes, paraffin oil, dog biscuits, machinery for the windmill, string, seeds, artificial manures; Trade with neighboring farms through a lawyer.
#'''What is ironic about the name of the solicitor Napoleon works with?'''
#*Mr. Whymper; They don't expect him to work but they expect him to be sneaky. 
#'''How do the animals feel about engaging in trade and using money with humans? How does Squealer convince them it is okay?'''
#*The animals don't like engaging in trade/using money with humans. They recall a resolution against it from the 1st meeting after the rebellion; the animal's disagreement is not written down anymore, so there is no such thing as their disagreement existing.
#'''What is the opinion of the town's people of Animal Farm and Mr. Jones?'''
#*Animal Farm - They hate the farm more than ever because it is successful, though they have respect for it by calling it Animal Farm.
#*Mr. Jones - Given up on him because he can't do it.
#'''Where have the pigs moved? How does Squealer explain this move to the animals?'''
#*Farmhouse; brains of farm need quiet places to work.
#'''What do the other animals discover about the pigs now living in the house?'''
#*They are sleeping on the beds.
#'''Clover asked Muriel to read her the fourth commandment on the barn wall -- what is different about this?'''
#*She adds &quot;with sheets&quot;... Squealer has added this new modification to the commandment. 
#'''How does Squealer once again manage to convince the animals about sleeping in the beds?'''
#*Bed: place to sleep, sheets are the human invention which is bad.
#'''What happens one night during a storm?'''
#*Windmill is destroyed.
#'''Who does Napoleon blame for this? What does he offer as a reward for his capture? What do the animals find when they are looking around the farm?'''
#*Snowball; a full bushel of apples and &quot;Animal Hero, Second Class&quot;; Footprints of a pig in the grass close to the knoll---Napoleon pronounced these footprints to be Snowball who probably came from Foxwood.

==Chapter 7==
#'''Based on the first two pages of the chapter, what is the animals' main concern?'''
#*Food supply while rebuilding the windmill.
#'''How does Napoleon try to fool Mr. Whymper?'''
#*Napoleon orders the mostly empty food bin to be filled with sand to the brim, then the remaining food placed on top, giving Mr. Whymper the allusion that all is fine on the farm.
#'''What is the agreement Napoleon makes with Mr. Whymper?'''
#*They will sell 400 eggs a week to a local grocer in order to buy more grain.
#'''Describe the reaction of the hens and how they try to get Napoleon to reconsider. What happens?'''
#*In protest to disliking it, they fig into rafters and lay their eggs, consequently, the eggs fall to the ground and break. Napoleon, angry, starves the 9 hens in 5 days.
#'''Why does Napoleon use Snowball in regards to selling the timber?'''
#*Napoleon says the different forms are hiding Snowball as a way o justify on-going negotiations in price. 
#'''Explain the use of Snowball as a scapegoat on the farm'''.
#*Scapegoat - Someone who is blamed for something that he/she has not done. Snowball is blamed whenever something goes wrong. 
#'''How does Squealer manage to change the animals' memories of Snowball during the battles and his true allegiance?'''
#*Through persuasive paragraphs of explanations (he explains that Snowball left the Battle of Cowshed at the critical moment, and not as a battle move; Snowball's bleeding was fake and it was just a coverup as Mr. Jone's agent)--he also uses Napoleon as an excuse (Napoleon said it himself!)--Professing that Napoleon is the one who did Snowball's actions in the battle.
#'''What happens to the animals who &quot;confess&quot; to being in league with Snowball or commit crimes in the eyes of the other animals?'''
#*They were killed right on the stop by the dogs.
#'''How has the animals' idea and realization of Animal Farm changed from the start of the book?'''
#*The Dream of Animal Farm - No hunger/whip no longer used, equal, everyone working to their best, strong protecting the weak.
#*Reality of Animal Farm - Growling and threatening dogs, animals being killed, Napoleon owns absolute monarchy for the farm.
#'''What have the pigs changed at the end of the chapter? What might this signify to the animals?'''
#*&quot;Beasts of England&quot; is forbidden because it is no longer needed since the rebellion is over and has already taken place; Gives them a sense of hopelessness, shows Napoleon is the absolute ruler of the farm.

==Chapter 8==
#'''By the end of the chapter, which two commandments have been altered? What do they now say? Why has each of them been changed?'''
#*No animal shall kill any other animal ''without cause''.
#*No animal shall sleep in a bed ''with sheets''.
#*To justify their actions
#'''What is Napoleon's new title? How has treatment of him changed in the first few pages?'''
#*&quot;Our Leader, Comrade Napoleon&quot;; He is rarely seen more often than once every 2 weeks... when he goes to places, a black roaster walks in front of him crowing. He now eats and sleeps alone away from the pigs.
#'''Explain why the animals do not like Frederick'''.
#*Because he treats his animals with cruelty and they want to liberate his farm.

#'''How does Napoleon double cross Pilkington?'''
#*Napoleon acts like he will sell the wood to Pilkington, but instead, he sells it to Frederick. 
#'''How is Napoleon double-crossed by Frederick? When Napoleon solicits Pilkington for help, what is the response he receives?'''
#*Napoleon is given fake money by Frederick in exchange for the timber. When Frederick and his men attack Animal Farm, he asks Pilkington for help in which he receives the response, &quot;serves you right&quot;.
#'''What events occur during the Battle of the Windmill?'''
#*The men attack the animals with guns while they're preparing the windmill. They blow up the windmill with dynamite which causes the animals to attack wildly. The men run out and the animals win. 
#'''How do the pigs manage to turn the misfortune into a victory? What does Napoleon reward himself with?'''
#*The victory was protecting the farm from Frederick's attack; his award: The Order of the Green Banner.
#'''To what realization does Boxer arrive about himself?'''
#*He's not as strong and tough as he used to be, he is now 11... it will be harder to build the windmill.
#'''Explain what happens when the pigs find the whiskey in the basement'''.
#*They drank it all before Napoleon bans alcohol.
#'''What occurs at the end of the chapter that only Benjamin fully understands?'''
#*In the middle of the night, the animals hear a loud sound. They discover Squealer, a broken ladder and a paintbrush. Only Benjamin understands that Squealer has been changing the commandments.

{{center top}}'''1. &quot;No animal shall drink alcohol TO EXCESS&quot;''' - This allows them to make alcohol!{{center bottom}}
==Chapter 9==
#'''Why are the animals so easily convinced that their situation is better than when Jones was around when it really isn't?'''
#*Squealer provides &quot;arguments&quot; for this, like &quot;more oats, more hay, etc.&quot;, worked shorter hours, better water lived longer, more infants surviving and fewer fleas. They, according to Squealer, were also &quot;free&quot;. This is easily accepted because no one remembers how life was like before the freedom from Mr. Jones. They also believe that now the animals are finally controlling the farm. 
#'''Once the piglets are born, how do the pigs further separate themselves from the other animals?'''
#*The piglets were turned over to be taught by Napoleon himself. Also, several rules were created, such as the piglets are discouraged from playing with the other animals, other animals have to step aside when pigs pass by, pigs have to wear special green ribbons on Sunday and all the barely on the farm were to be left for the pigs.
#'''What is the purpose of the &quot;Spontaneous Demonstrations&quot;?'''
#*To celebrate the struggles and triumphs of Animal Farm, truth: Glorify Napoleon and distract them from their hardships. 
#'''How does the farm change once it becomes the month of April?'''
#*The farm is now a Republic and Napoleon is the president of it. 
#'''Why has Moses suddenly appeared? Why do the pigs allow him to stay on the farm?'''
#*To tell the animals that behind a dark cloud is Sugar Candy Mountain; They allow him on the farm as long as Moses gives a gill of beer a day. 
#'''Give the chain of events from Boxer's injury to his leaving Animal Farm'''.
#*Boxer's hoof is injured during the Battle of the Windmill, so after his lungs give out and he collapses while working, Boxer receives medicine from Clover for the next two days until he was taken away to his death (a &quot;hospital&quot; in Willingdon in care of humans).
#'''How does Squealer continue to lie to the animals about the reality of Boxer's death?'''
#*He says that Boxer was taken to the hospital. The van Boxer was taken in was previously the property of the knacker.
#'''What occurs at the end of the chapter which demonstrates the real reason for Boxer's death?'''
#*Reason for Boxer's death was to sell him to a knacker in order to make money to buy whiskey.

==Chapter 10==
#'''In the first three paragraphs of the chapter, how are the farm and animals different?''' 
#*Benjamin: Sad and grumpy
#*Squealer: Fat
#*Napoleon: Fat
#*No one remembers Snowball, Boxer.
#*Muriel, Bluebell, Jessie, and Pincher are dead.
#*Clover: Old/2 years past retirement.
#*More creatures on the farm. 
#*Rebellion is pretty much forgotten and the windmill is used for coal and harvesting. 
#'''According to Napoleon, what is the animals' truest happiness?'''
#*Working hard and living frugally.
#'''In relation to work, how have the pigs and dogs come to resemble the humans at the start of the chapter?'''
#*Pigs and dogs are starting to consume large amounts of food without producing.
#'''Where has Squealer taken the sheep? What is the new maxim?'''
#*Another side of the farm; &quot;Four legs good, 2 legs better&quot;.
#'''What events have startled first Colver, then the other animals?'''.
#*The pigs walked on their hind legs (walking on two legs).
#'''What item does Napoleon possess now?'''
#*A whip.
#'''What has happened to the commandments at the end of the barn?'''
#*They covered the other commandments with tar and is replaced with &quot;All animals are equal, but some animals are more equal than others&quot;. Why?: To justify all their actions.
#'''Why does Benjamin break his rule to never read the commandments?'''
#*They, him and Clover, are pretty much the only animals alive from the start of the novels and he doesn't want the truth to be kept from her now that the pigs walk on 2 legs. 
#'''What behaviors of the pigs have the other animals accepted without question because of the new commandment?'''
#*They started to wear the Jones's family clothes.
#*Pigs carrying whips.
#*Napoleon is smoking pipe.
#*Bought themselves a radio.
#*Newspapers (Daily Mirror).
#'''What sight do the animals see through the dining room window?'''
#*Animals sitting at a table playing cards and drinking.
#'''Summarize Pilkington's toast.'''
#*Order of the farm and the organization of it through fear and whips. The farm is also productive. 
#'''What changes has Napoleon told the humans he is making to the farm?'''
#*Manor Farm as the new name.
#*No more &quot;comrade&quot;.
#*No more walking past a boar's skull; because it reflects Animalism and the rebellion. 
#*Changes flag to plain green flag.
#*Basically, Napoleon is becoming Mr. Jones.
#'''Look at the last incident of the story in the last two paragraphs. Explain what happens and in your own words explain the last paragraph.'''
#*Mr. Pilkington and Napoleon both play the ace of spades showing they're both cheaters. They're both cheaters and can't be trusted, which is why the animals couldn't tell the difference between the animals an humans. They are similar.


[[Category:Books]]</text>
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      <text bytes="24384" xml:space="preserve">==Characters==
*'''Mr. Jones''' - Czar Nicholas II
*'''Old Major''' - Karl Max
*'''Snowball''' - Trotsky
*'''Napoleon''' - Stalin
*'''Squealer''' - Stalin's propoganda committee
*'''Dogs''' - KGB
*'''Boxer''' - dedicated but tricked communist supporters
*'''Moses''' - religion
*'''Frederick''' - Hitler/Deutschland
*'''Pilkington''' - FDR/Churchill/Allie Forces
*'''Whymper''' - Capitalist Westerners 

==Events/Objects==
*'''Windmill''' - 5 year plans
*'''Major's Skull''' - Lenin's body
*'''Battle of the Cowshed''' - Russian Revolution
*'''Animal Executions''' - Stalin's purge
*'''Battle of the Windmill''' - WWII

==Chapter 1==
#'''For what purpose did Old Major call the meeting of the animals?'''
#*To share a dream he had.
#'''According to Major, what is the cause of all the animals' problems? Be specific.'''
#*Mankind; Man consumes without producing and only gives the bare minimum to prevent starving. He kills animals who are old and useless, they are selfish also. Sells the young offspring for profit. Humans don't know how to best use the land.
#'''After they vote and decide rats are comrades, Major summarizes his points for the animals to remember. What are they?'''
#*2 legs - enemy 
#*4 legs or wings - friend
#*Don't resemble a man, like living in a house, sleeping on a bed, touch money, etc. (no alcohol, trade, no clothes, smoke). 
#*Treat each other equally and with no tyranny.

==Chapter 2==
#'''Who gained leadership of the animals? Why?'''
#*Pigs; Most clever animals on the farm.
#'''What concept did the three pigs come up with?'''
#*Animalism
#'''How did the other animals respond to this concept? What are some of the questions the animals had for the pigs?'''
#*The other animals did not like it or understand because they did not think it would happen in their lifetimes. They want to know who will feed us after Mr. Jones is gone and why should we care about what happens after we die. Why should we work for it if it will happen anyways?
#'''Who is Moses? What idea does he spread around to the animals?'''
#*Tame Raven - Mr. Jones's pet; Religion (sugar candy mountain)
#'''What impact do Clover and Boxer have on others?'''
#*Persuaded other animals to believe in Animalism through simple arguments. 
#'''Who gained leadership of the animals? Why?'''
#*Pigs; Most cleverest animals on the farm.
#'''What concept did the three pigs come up with?'''
#*Animalism
#'''How did the other animals respond to this concept? What are some of the questions the animals had for the pigs?'''
#*The other animals did not like it or understand because they did not think it would happen in their lifetimes. They want to know who will feed us after Mr. Jones is gone and why should we care about what happens after we die. Why should we work for it if it will happen anyway?
#'''Who is Moses? What idea does he spread around to the animals?'''
#*Tame Raven--Mr. Jones's pet; religion (sugar candy mountain)
#'''What impact do Clover and Boxer have on others?'''
#*Persuaded other animals to believe in Animalism through simple arguments.
#'''What events lead up to the rebellion?'''
#*Mr. Jones started to neglect the farm, not feeding the animals, men were idle and dishonest, fields full with weed, buildings needed roofing, hedges were neglected.
#'''What do the animals do that causes the rebellion to happen?'''
#*One of the cows broke into the store and the animals began to help themselves to the bins, waking Mr. Jones up. 
#'''Describe the events that happen in the rebellion from beginning to end.'''
#*Mr. Jones wakes up. He and his men begin to lash their whips frantically.
#*The animals retaliate against the whips by jumping on them. They start to butt and kick Mr. Jones and his men. This causes Mr. Jones and men to leave the farm on foot 2 minutes later.
#*Mrs. Jones is chased by Moses.
#*Mr. Jones and his men are defeated and the Manor Farm is theirs.
#'''What are the first four things that the animals do once they win the rebellion?'''
#*Checked to make sure no human remains on the farm.
#*Throw out remains that was used/a sign of Mr. Jones's terror, like knives.
#*Napoleon leads the animals back to the store and gives them corn and 2 biscuits for the dogs.
#*They sing Beasts of England and then settled down and slept for the night.
#'''Where is Mollie discovered to be when the animals go through the house?'''
#*Mollie was behind in the best bedroom admiring herself.
#'''What have the pigs taught themselves to do? To what do they change the name of the farm to?'''
#*The pigs have taught themselves to read and write from a spelling book from Mr. Jones; The name was changed to Animal Farm.
#'''List the 7 commandments of Animalism'''
#*Whatever goes upon two legs is an enemy.
#*Whatever goes upon four legs or has wings is a friend.
#*No animal shall wear clothes.
#*No animal shall sleep in a bed.
#*No animal shall drink alcohol.
#*No animal shall kill any other animal.
#*All animals are equal.
#'''What happens to the milk?'''
#*The milk disappears.

==Chapter 3==
#'''What is different about the harvest this year? How does this represent the true idea of Communism?'''
#*The harvest was a lot more successful than with Mr. Jones and his men; Production of the farm truly belongs to the animals.
#'''Describe Boxer at the beginning of the chapter. What is his motto?'''
#*Boxer worked even harder than he worked when he was working for Mr. Jones. He is the admiration for everybody; &quot;I will work harder&quot;.
#'''How have the animals changed at the start of the chapter with their newly found freedom? Who hasn't changed?'''
#*The animals now truly love eating their food knowing that they produced it themselves without their cruel masters giving them the bare minimum. The animals work together as a team and barely argue; Mollie hasn't changed because she only cares about herself.
#'''What routine do the animal engage in on Sunday mornings? Explain the flag and its elements.'''
#*They don't do any work on Sunday and breakfast was an hour later than usual. A ceremony takes place after breakfast; The flag was an old green tablecloth painted with a hoof and horn. The hoof and horn signified the Republic of the Animals. 
#'''What is the degree of success achieved by other animals on the farm when it comes to reading?'''
#*Almost every animal on the farm was literate in some degrees by the autumn as reading an writing classes which were held at the farm were a success; Mollie only knows the letters of her name, Boxer couldn't get past the letter &quot;D&quot;, pigs knew the alphabets, dogs were behind the pigs. 
#'''What maxim was created for the farm? Why?'''
#*The maxim that was created was &quot;Four legs good, two legs bad&quot; and it was created because some of the animals (sheep, hens, ducks) could not learn the 7 commandments.
#'''Where does Napoleon take Jessie and Bluebell's pups? What do you think he is doing with them?'''
#*Napoleon took them to a loft which could only be reached by a ladder from the Larness-room; teaching them by himself.
#'''How does Squealer explain the disappearance of the milk and the apples?'''
#*He explains that the pigs need the milk and apples more as they need the brian-helping nutrients from the milk and apples more than the other animals as they're the ones who rule the other animals.
#'''What is the main argument Squealer uses to ensure what he says is taken seriously by the other animals?'''
#*The main argument: &quot;If the pigs fails, then Mr. Jones will be back!&quot;

==Chapter 4==
#'''Contrast Mr. Pilkington to Mr. Frederick'''.
#*'''Mr. Pilkington''': Represents Allied Forces, easy-going, hunted or fish by the seas, owns an unorganized farm: Foxwood.
#*'''Mr. Frederick''': Represents Germany, tough-shrewd, involved in many lawsuits, hard bargains, owns an organized farm: Pinchfield. 
#'''How do the humans respond to their own animals singing &quot;Beasts of England&quot;?'''
#*The animals are whipped on the spot if they are heard singing it.
#'''Why does no one go to help Mr. Jones reclaim his farm at the beginning of the chapter?'''
#*Everyone is secretly trying to figure out a way to benefit from Mr. Jones' bad situation.
#'''Who led the first counter-attack when Mr. Jones returned to the farm with other men to try to reclaim it? What influenced him to lead the counter-attack?'''
#*Snowball leads the counter-attack after he reads a book on Julius Cesear's military experiences. 
#'''How do the animals trick Mr. Jones and the other men?'''
#*The animals run back into the farmyard acting as if they are retreating, but they are leading the men into an ambush.
#'''What happens to Snowball? The sheep? Mollie?'''
#*Mr. Jones fires his gun at Snowball, but the pellets graze his back and kill the sheep behind him. Mollie is hiding in her stall instead of fighting.
#'''How does Boxer react to the idea of killing a human?'''
#*Boxer is saddened at the thought of killing someone, even a human. 
#'''What awards do Snowball and Boxer receive? The sheep?'''
#*Snowball + Boxer = &quot;Animal Hero, First Class&quot;
#*Dead sheep = &quot;Animal Hero, Second Class&quot;
#'''What do they do with Mr. Jones's gun?'''
#*The gun is placed at the bottom of the flagpole. It will be fired twice a year once on the anniversary of Battle of Cowshed on October 12 and Midsummer's Day for the rebellion.

==Chapter 5==
#'''What has Clover discovered about Mollie? What has she found in Mollie's stall?'''
#*Clover has found out that Mollie is interacting with the humans and Clover finds her 2 favorite things: Lump of sugar and ribbons. 
#'''What happened to Mollie? How is that treated by other animals?'''
#*Mollie runs away to joy to another farm across town so she can have sugar, wear ribbons and be a pet; They animals never speak fo Mollie against because she betrayed them. 
#'''What idea does Snowball come up with for the farm? What does he hope that it will provide for the farm? How will it make their lives better?'''
#*Windmill; Electricity created will power machines as well as produce light and heat for each stall; They will benefit from more comfort and less work.
#'''What is Napoleon's opinion of Snowball's plan? What does he do to show it?'''
#*Napoleon is not impressed by his plans; Urinates on it.
#'''During a Sunday meeting, the animals are trying to decide whether or not to go ahead and build the windmill. What is Snowball's argument? Napoleon's?'''
#*Snowball claims it will lower their work week to 3 days a week.
#*Napoleon believes its nonsense.
#'''What happens when the animals decide to go with Snowball's plan?'''
#*Napoleon has the dogs, KGB, run Snowball off the farm.
#'''What is the first decision that Napoleon makes now that he is in charge?'''
#*He abolishes the Sunday morning meetings.
#'''How will decisions be made on the farm?'''
#*By a special committee of pigs presided over by Napoleon himself.
#'''What new step has been added to the Sunday morning processional?'''
#*Animals must walk past the skull of Old Major that has been dug up and placed at the bottom of the flagpole next to Mr. Jones's gun.
#'''How has the seating arrangement changed during the meetings?'''
#*Napoleon, Squealer, and a pig named Minimus sat together on the raised platform like Old Major used to do. The dogs sit in front of the three of them and the rest of the pigs are on the back end of the platform. The separation of the pigs and dogs is indicating that their perceived superiority to the other animals.
#'''How does Napoleon use the sheep to help quiet arguments that oppose his plans?'''
#*The sheep yell repeatedly, &quot;4 legs good, 2 legs bad&quot;, over-top of anyone trying to rebel.
#'''What does Napoleon decide to do after all? How does Squealer convince the animals that Napoleon supported the plan all along?'''
#*Napoleon decides to build the windmill; He tells that it was Napoleon's idea all along and the growling of the dogs keeps the animals from denying the idea.

==Chapter 6==
#'''Reread the opening sentence of this chapter. What do you notice is significant in the diction?'''
#*The animals are working like slaves... this is the same way they were treated with Mr. Jones.
#'''What changes does Napoleon make to the work week?'''
#*They work 60 hours a week (10 hours a day) - now work on Sundays (they say it is voluntary, but if an animal does not work on Sundays, his/her food will be cut in half).
#'''What items become scarce that the farm cannot produce? How does Napoleon decide they will overcome these hardships?'''
#*Nails, iron for horseshoes, paraffin oil, dog biscuits, machinery for the windmill, string, seeds, artificial manures; Trade with neighboring farms through a lawyer.
#'''What is ironic about the name of the solicitor Napoleon works with?'''
#*Mr. Whymper; They don't expect him to work but they expect him to be sneaky. 
#'''How do the animals feel about engaging in trade and using money with humans? How does Squealer convince them it is okay?'''
#*The animals don't like engaging in trade/using money with humans. They recall a resolution against it from the 1st meeting after the rebellion; the animal's disagreement is not written down anymore, so there is no such thing as their disagreement existing.
#'''What is the opinion of the town's people of Animal Farm and Mr. Jones?'''
#*Animal Farm - They hate the farm more than ever because it is successful, though they have respect for it by calling it Animal Farm.
#*Mr. Jones - Given up on him because he can't do it.
#'''Where have the pigs moved? How does Squealer explain this move to the animals?'''
#*Farmhouse; brains of farm need quiet places to work.
#'''What do the other animals discover about the pigs now living in the house?'''
#*They are sleeping on the beds.
#'''Clover asked Muriel to read her the fourth commandment on the barn wall -- what is different about this?'''
#*She adds &quot;with sheets&quot;... Squealer has added this new modification to the commandment. 
#'''How does Squealer once again manage to convince the animals about sleeping in the beds?'''
#*Bed: place to sleep, sheets are the human invention which is bad.
#'''What happens one night during a storm?'''
#*Windmill is destroyed.
#'''Who does Napoleon blame for this? What does he offer as a reward for his capture? What do the animals find when they are looking around the farm?'''
#*Snowball; a full bushel of apples and &quot;Animal Hero, Second Class&quot;; Footprints of a pig in the grass close to the knoll---Napoleon pronounced these footprints to be Snowball who probably came from Foxwood.

==Chapter 7==
#'''Based on the first two pages of the chapter, what is the animals' main concern?'''
#*Food supply while rebuilding the windmill.
#'''How does Napoleon try to fool Mr. Whymper?'''
#*Napoleon orders the mostly empty food bin to be filled with sand to the brim, then the remaining food placed on top, giving Mr. Whymper the allusion that all is fine on the farm.
#'''What is the agreement Napoleon makes with Mr. Whymper?'''
#*They will sell 400 eggs a week to a local grocer in order to buy more grain.
#'''Describe the reaction of the hens and how they try to get Napoleon to reconsider. What happens?'''
#*In protest to disliking it, they fig into rafters and lay their eggs, consequently, the eggs fall to the ground and break. Napoleon, angry, starves the 9 hens in 5 days.
#'''Why does Napoleon use Snowball in regards to selling the timber?'''
#*Napoleon says the different forms are hiding Snowball as a way o justify on-going negotiations in price. 
#'''Explain the use of Snowball as a scapegoat on the farm'''.
#*Scapegoat - Someone who is blamed for something that he/she has not done. Snowball is blamed whenever something goes wrong. 
#'''How does Squealer manage to change the animals' memories of Snowball during the battles and his true allegiance?'''
#*Through persuasive paragraphs of explanations (he explains that Snowball left the Battle of Cowshed at the critical moment, and not as a battle move; Snowball's bleeding was fake and it was just a coverup as Mr. Jone's agent)--he also uses Napoleon as an excuse (Napoleon said it himself!)--Professing that Napoleon is the one who did Snowball's actions in the battle.
#'''What happens to the animals who &quot;confess&quot; to being in league with Snowball or commit crimes in the eyes of the other animals?'''
#*They were killed right on the stop by the dogs.
#'''How has the animals' idea and realization of Animal Farm changed from the start of the book?'''
#*The Dream of Animal Farm - No hunger/whip no longer used, equal, everyone working to their best, strong protecting the weak.
#*Reality of Animal Farm - Growling and threatening dogs, animals being killed, Napoleon owns absolute monarchy for the farm.
#'''What have the pigs changed at the end of the chapter? What might this signify to the animals?'''
#*&quot;Beasts of England&quot; is forbidden because it is no longer needed since the rebellion is over and has already taken place; Gives them a sense of hopelessness, shows Napoleon is the absolute ruler of the farm.

==Chapter 8==
#'''By the end of the chapter, which two commandments have been altered? What do they now say? Why has each of them been changed?'''
#*No animal shall kill any other animal ''without cause''.
#*No animal shall sleep in a bed ''with sheets''.
#*To justify their actions
#'''What is Napoleon's new title? How has treatment of him changed in the first few pages?'''
#*&quot;Our Leader, Comrade Napoleon&quot;; He is rarely seen more often than once every 2 weeks... when he goes to places, a black roaster walks in front of him crowing. He now eats and sleeps alone away from the pigs.
#'''Explain why the animals do not like Frederick'''.
#*Because he treats his animals with cruelty and they want to liberate his farm.

#'''How does Napoleon double cross Pilkington?'''
#*Napoleon acts like he will sell the wood to Pilkington, but instead, he sells it to Frederick. 
#'''How is Napoleon double-crossed by Frederick? When Napoleon solicits Pilkington for help, what is the response he receives?'''
#*Napoleon is given fake money by Frederick in exchange for the timber. When Frederick and his men attack Animal Farm, he asks Pilkington for help in which he receives the response, &quot;serves you right&quot;.
#'''What events occur during the Battle of the Windmill?'''
#*The men attack the animals with guns while they're preparing the windmill. They blow up the windmill with dynamite which causes the animals to attack wildly. The men run out and the animals win. 
#'''How do the pigs manage to turn the misfortune into a victory? What does Napoleon reward himself with?'''
#*The victory was protecting the farm from Frederick's attack; his award: The Order of the Green Banner.
#'''To what realization does Boxer arrive about himself?'''
#*He's not as strong and tough as he used to be, he is now 11... it will be harder to build the windmill.
#'''Explain what happens when the pigs find the whiskey in the basement'''.
#*They drank it all before Napoleon bans alcohol.
#'''What occurs at the end of the chapter that only Benjamin fully understands?'''
#*In the middle of the night, the animals hear a loud sound. They discover Squealer, a broken ladder and a paintbrush. Only Benjamin understands that Squealer has been changing the commandments.

{{center top}}'''1. &quot;No animal shall drink alcohol TO EXCESS&quot;''' - This allows them to make alcohol!{{center bottom}}
==Chapter 9==
#'''Why are the animals so easily convinced that their situation is better than when Jones was around when it really isn't?'''
#*Squealer provides &quot;arguments&quot; for this, like &quot;more oats, more hay, etc.&quot;, worked shorter hours, better water lived longer, more infants surviving and fewer fleas. They, according to Squealer, were also &quot;free&quot;. This is easily accepted because no one remembers how life was like before the freedom from Mr. Jones. They also believe that now the animals are finally controlling the farm. 
#'''Once the piglets are born, how do the pigs further separate themselves from the other animals?'''
#*The piglets were turned over to be taught by Napoleon himself. Also, several rules were created, such as the piglets are discouraged from playing with the other animals, other animals have to step aside when pigs pass by, pigs have to wear special green ribbons on Sunday and all the barely on the farm were to be left for the pigs.
#'''What is the purpose of the &quot;Spontaneous Demonstrations&quot;?'''
#*To celebrate the struggles and triumphs of Animal Farm, truth: Glorify Napoleon and distract them from their hardships. 
#'''How does the farm change once it becomes the month of April?'''
#*The farm is now a Republic and Napoleon is the president of it. 
#'''Why has Moses suddenly appeared? Why do the pigs allow him to stay on the farm?'''
#*To tell the animals that behind a dark cloud is Sugar Candy Mountain; They allow him on the farm as long as Moses gives a gill of beer a day. 
#'''Give the chain of events from Boxer's injury to his leaving Animal Farm'''.
#*Boxer's hoof is injured during the Battle of the Windmill, so after his lungs give out and he collapses while working, Boxer receives medicine from Clover for the next two days until he was taken away to his death (a &quot;hospital&quot; in Willingdon in care of humans).
#'''How does Squealer continue to lie to the animals about the reality of Boxer's death?'''
#*He says that Boxer was taken to the hospital. The van Boxer was taken in was previously the property of the knacker.
#'''What occurs at the end of the chapter which demonstrates the real reason for Boxer's death?'''
#*Reason for Boxer's death was to sell him to a knacker in order to make money to buy whiskey.

==Chapter 10==
#'''In the first three paragraphs of the chapter, how are the farm and animals different?''' 
#*Benjamin: Sad and grumpy
#*Squealer: Fat
#*Napoleon: Fat
#*No one remembers Snowball, Boxer.
#*Muriel, Bluebell, Jessie, and Pincher are dead.
#*Clover: Old/2 years past retirement.
#*More creatures on the farm. 
#*Rebellion is pretty much forgotten and the windmill is used for coal and harvesting. 
#'''According to Napoleon, what is the animals' truest happiness?'''
#*Working hard and living frugally.
#'''In relation to work, how have the pigs and dogs come to resemble the humans at the start of the chapter?'''
#*Pigs and dogs are starting to consume large amounts of food without producing.
#'''Where has Squealer taken the sheep? What is the new maxim?'''
#*Another side of the farm; &quot;Four legs good, 2 legs better&quot;.
#'''What events have startled first Colver, then the other animals?'''.
#*The pigs walked on their hind legs (walking on two legs).
#'''What item does Napoleon possess now?'''
#*A whip.
#'''What has happened to the commandments at the end of the barn?'''
#*They covered the other commandments with tar and is replaced with &quot;All animals are equal, but some animals are more equal than others&quot;. Why?: To justify all their actions.
#'''Why does Benjamin break his rule to never read the commandments?'''
#*They, him and Clover, are pretty much the only animals alive from the start of the novels and he doesn't want the truth to be kept from her now that the pigs walk on 2 legs. 
#'''What behaviors of the pigs have the other animals accepted without question because of the new commandment?'''
#*They started to wear the Jones's family clothes.
#*Pigs carrying whips.
#*Napoleon is smoking pipe.
#*Bought themselves a radio.
#*Newspapers (Daily Mirror).
#'''What sight do the animals see through the dining room window?'''
#*Animals sitting at a table playing cards and drinking.
#'''Summarize Pilkington's toast.'''
#*Order of the farm and the organization of it through fear and whips. The farm is also productive. 
#'''What changes has Napoleon told the humans he is making to the farm?'''
#*Manor Farm as the new name.
#*No more &quot;comrade&quot;.
#*No more walking past a boar's skull; because it reflects Animalism and the rebellion. 
#*Changes flag to plain green flag.
#*Basically, Napoleon is becoming Mr. Jones.
#'''Look at the last incident of the story in the last two paragraphs. Explain what happens and in your own words explain the last paragraph.'''
#*Mr. Pilkington and Napoleon both play the ace of spades showing they're both cheaters. They're both cheaters and can't be trusted, which is why the animals couldn't tell the difference between the animals an humans. They are similar.


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 | name          = Low Danish
 | nativename    = ''Loxdanš''
 | pronunciation = loxdanʃ
 | states        = Germanic world
 |image          = 
 | imagecaption   = 
 | speakers      = unknown
 | ethnicity= 
 | date          = 2018
 | ref           = e18
 | familycolor   = Indo-European
 | fam2          = [[Germanic languages|Germanic]]
 | fam3          = 
 | script        = [[Latin script]]: &lt;br&gt; Low Danish Alphabet, &lt;br&gt; International Low Danish alphabet.
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}} 
{{languages}}
&lt;div style=&quot;color: blue; font-size: 200%;&quot;&gt;{{center|'''Velonkomneron!'''}}&lt;/div&gt;
{{center|[[File:Bonnie Blue flag.svg|95px]]}}

Welcome to the '''Department of Low Danish''', the Unified Germanic Language, at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language Studies|School of Language Studies]].

''This language is under grammatical change.''
== Introduction ==
Loxdanš is a recently created language which seeks to unify the Germanic communities of England, Germany, Netherlands, Scandinavia and The Northern Germanic communities.

Learning Loxdanš or Low Danish is also a way to know other Germanic languages easily without working on them seperately.

== What is Low Danish? ==
Low Danish is a Germanic conlang, inspired by English, German, Dutch and Danish, with minor Persian and Slavic inspiration. With it’s very simple grammar and lexicon, it is easy to learn it. Feel free to learn it, it can be useful at some point!

Low Danish is created by a team, called the Low Danish team. To communicate with the team, send a message to [[user:Hifnakia|Hifnakia]], as he is a member of the team.

== Stories ==
=== Jakobšon Family ===
* [[Loxdanš/Joxjen|Joxjen]]
* [[Loxdanš/Mërij|Mërij]]
* [[Loxdanš/Market|Market]]
=== Historical events ===
* [[Loxdanš/Berlin Vë|Berlin Vë]]
* [[Loxdanš/Sovijetbina|Sovijetbina]]
* [[Loxdanš/3. Krëg|3. Krëg]]

=== Bible stories ===

== Courses ==

* {{75Percent}} [[Loxdanš/Lesson 1|Lesson 1]] - Pronunciation, Alphabet and noun basics.
* {{75Percent}} [[Loxdanš/Lesson 2|Lesson 2]] - Repetition of Lesson 1, Verb and adjective basics and greetings.
* {{100Percent}} [[Loxdanš/Lesson 3|Lesson 3]] - Adverbs, endings -te and -ten
* {{50Percent}} [[Loxdanš/Lesson 4|Lesson 4]] - Basic vocabulary
* {{50Percent}} [[Loxdanš/Lesson 5|Lesson 5]] - Intermediate vocabulary.
* {{50Percent}} [[Loxdanš/Lesson 6|Lesson 6]] - Theme: Family
* {{50Percent}} [[Loxdanš/Lesson 7|Lesson 7]] - Theme: Countries
* {{50Percent}} [[Loxdanš/Lesson 8|Lesson 8]] - Theme: Nature
* {{25Percent}} [[Loxdanš/Lesson 9|Lesson 9]] - Theme: Food and Everyday things
* {{0Percent}} [[Loxdanš/Lesson 10|Lesson 10]] - Theme: How to read news and the whole world
* {{0Percent}} [[Loxdanš/Lesson 11|Lesson 11]] - Repetition and the rest of Loxdanš!

== Contributers ==

[[User:Hifnakia|Hifnakia]] ([[User talk:Hifnakia|discuss]] • [[Special:Contributions/Hifnakia|contribs]])</text>
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      <comment>Removing [[:c:File:2015_Mayhem_Festival_.jpg|2015_Mayhem_Festival_.jpg]], it has been deleted from Commons by [[:c:User:Pi.1415926535|Pi.1415926535]] because: [[:c:COM:DW|Derivative work]] of non-free content ([[:c:COM:CSD#F3|F3]]): content was: &quot;== {{int</comment>
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      <text bytes="2719" xml:space="preserve">[https://en.wikipedia.org/wiki/Rockstar_(drink) Rockstar energy] [https://en.wikipedia.org/wiki/Mayhem_Festival Mayhem fest] was a [https://en.wikipedia.org/wiki/Heavy_metal_music Heavy metal] tour that ran from 2008 till 2015. It featured big name bands such as [https://en.wikipedia.org/wiki/Slipknot_(band) Slipknot],[https://en.wikipedia.org/wiki/Amon_Amarth Amon Amarth], and [https://en.wikipedia.org/wiki/King_Diamond_(band) King Diamond] This is one of the few concert tours that has branched out from the United states to Canada which is usually a &quot;dead zone&quot; for tours and concert series. It was also a marketing ploy where Rockstar Energy would sample their newest products as unlimited free samples. Many of the sponsors would also sell and sample there products and many small buisnesses would be there to promote themselves.
 

Stages and events 

Each year the [https://en.wikipedia.org/wiki/Metal_Mulisha Metal Mulisha] would have a separate motorcycle stunt show. The bands would also have their own meet and greet, if they so desire. Over the years, the sponsor stage has gone through a few name changes. It started as the [https://en.wikipedia.org/wiki/Hot_Topic Hot Topic] stage (2008-2009), it then went to The Silver Star Stage in 2010,It changed again in 2011 to The [https://en.wikipedia.org/wiki/Revolver_(magazine) Revolver] Stage. And finally from 2012 till 2013 [https://en.wikipedia.org/wiki/Sumerian_Records Sumerian Records] named the stage after themselves, Then finally the last two years were sponsered by [https://en.wikipedia.org/wiki/Victory_Records Victory Records].

Common locations
*Washington     
*California   
*Arizona      
*New Mexico   
*Colorado     
*Kansas
*Texas
*Ontario
*Florida
*New York

Reasons for ending 

One of the main reasons for the cancellation was that metal was belive to be a dying gnere and it wasnt getting the recognition it needed. Another reason was the fact that the money which once was wnough is no longer enought to pay for all the bands at all the venues, which lead to some stages being left empty.

 Another reason was co-founder [https://en.wikipedia.org/wiki/Kevin_Lyman Kevin Lyman] tweeted that &quot;Metal is fat,gray,and old&quot;. this created backlash and sparked a small uprising in the metal community.

[[File:KerryKingMayhemFestival.jpg|thumb|[[wikipedia:Kerry_King|Kerry King at 2009 Mayhem Fest]]]]


References

[http://www.metalinjection.net/latest-news/mayhem-fest-co-founder-confirms-2015-is-the-last-year-of-the-tour Co-Founder confirms 2015 is the last year for the festival]

[https://www.metalinsider.net/festivals/r-i-p-rockstar-energy-drink-mayhem-festival-2008-2015 R I P Mayhem Fest 2008-2015]

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== Licensing ==
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{{Information
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|Source=https://en.wikiversity.org/wiki/WikiJournal_Preprints/Orientia_tsutsugamushi,_the_agent_of_scrub_typhus
|Date=14 September 2018
|Author= Kholhring Lalchhandama 
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== Licensing ==
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{{Information
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== Licensing ==
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{{Non-free use rationale 2
|Description = Photograph of the painting ''Men in the Market Place, Tangier'' by Hilda Rix Nicholas
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|Date = c. 1912
|Author = Hilda Rix Nicholas
|Article = WikiJournal_Preprints/Hilda_Rix_Nicholas
|Purpose = To support encyclopedic discussion of this work in this article. The illustration is specifically needed to support the following point(s): &lt;br/&gt;
Demonstrating Rix Nicholas's Moroccan work, for which she first became famous, including her movement toward post-impressionism and orientalism, as outlined in the article
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|Minimality = The artist is deceased and thus could not benefit from the reproducton of the work; the image is too low quality (and of only part of a work) to support commercial reproduction.
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==Licensing==
{{Non-free 2D art|image has rationale=yes}}
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{{Non-free use rationale 2
|Description = Photograph of the painting ''In Australia'' by Hilda Rix Nicholas
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|Purpose = To support encyclopedic discussion of this work in this article. The illustration is specifically needed to support the following point(s): &lt;br/&gt;
The artist's typical subject, and representation of the Australian pastoral ideal
|Replaceability = There are no known free images of Rix Nicholas's works. 
|Minimality = The artist is deceased and thus could not benefit from the reproducton of the work; the image is too low quality to support commercial reproduction.
|Commercial = n.a.
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==Licensing==
{{Non-free 2D art|image has rationale=yes}}
{{Category ordered by date|Non-free files uploaded as object of commentary|2014|03|17}}
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{{Non-free use rationale 2
|Description = Photograph of the painting ''Les fleurs dédaignées'' by Hilda Rix Nicholas
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|Date = 1925
|Author = Hilda Rix Nicholas
|Article = WikiJournal_Preprints/Hilda_Rix_Nicholas
|Purpose = To support encyclopedic discussion of this work in this article. The illustration is specifically needed to support the following point(s): &lt;br/&gt;
The unusual subject, the scale of the artist's ambition, and the deviation from the artist's style in other major works
|Replaceability = There is no applicable [[freedom of panorama]] that might allow photography of an image on public display.
|Minimality = The artist is deceased and thus could not benefit from the reproduction of the work; the image is too low quality to support commercial reproduction.
|Commercial = n.a.
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==Licensing==
{{Non-free 2D art|image has rationale=yes}}
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{{Non-free use rationale 2
|Description = Photograph of the painting ''The Summer House'' by Hilda Rix Nicholas
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|Article = WikiJournal_Preprints/Hilda_Rix_Nicholas
|Purpose = To support encyclopedic discussion of this work in this article. The illustration is specifically needed to support the following point(s): &lt;br/&gt;
the stereotypical subject matter and the departure from the artist's prevailing depiction of landscape at that time
|Replaceability = There are no known free images of Rix Nicholas's works.
|Minimality = The artist is deceased and thus could not benefit from the reproducton of the work; the image is too low quality to support commercial reproduction.
|Commercial = n.a.
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==Licensing==
{{Non-free 2D art|image has rationale=yes}}
{{Category ordered by date|Non-free files uploaded as object of commentary|2014|03|17}}
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== License ==
{{PD}}
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{{PD-self}}
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{{Information Q|Q81434400}}

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{{Information
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== Licensing ==
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{{Information
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
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{{Information
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== Licensing ==
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{{Information
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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{{Information
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== Licensing ==
{{PD-self}}
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{{Information
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== Licensing ==
{{PD-self}}
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{{Information
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== Licensing ==
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== Licensing ==
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== Licensing ==
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      <text bytes="61495" xml:space="preserve"> https://www.youtube.com/watch?v=KIzN2tIynxM&amp;t=1292s

 [https://www-worldscientific-com.wikipedialibrary.idm.oclc.org/doi/epdf/10.1142/13332 &lt;span style=&quot;color:indigo; font-size: 77px;&quot;&gt; 〠 &lt;/span&gt;] &lt;-- Current read from WikiLibrary's worldscientific-com.wikipedialibrary.idm.oclc.org 
 [https://ja.wikipedia.org/wiki/メインページ &lt;span style=&quot;color:indigo; font-size: 33px;&quot;&gt; ja.wikipedia.org/wiki/メインページ &lt;/span&gt;]
Works in Progress: thestory@kgw.com 503 226 5090; inVincible [https://wikipedialibrary.wmflabs.org/?markasread=423128&amp;markasreadwiki=enwikiversity library]
{|
|-
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff&quot;&gt; 0 &lt;/span&gt;
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff&quot;&gt; ◯ &lt;/span&gt;
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff&quot;&gt; ∅  &lt;/span&gt;
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff&quot;&gt;  2030 February 28th&lt;/span&gt; 
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff&quot;&gt;  〒 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 1 &lt;/span&gt;
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 一 &lt;/span&gt;
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt;  ^ &lt;/span&gt;
|| 2028 February 29th || 〒 272 
 
|-
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 2 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt;  二 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; | &lt;/span&gt;
|| 2024 October 31st || 〠〒 97305 || passive over winter

|-
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 3 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt;  三 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt;  . &lt;/span&gt;
|| 2024 July 23rd

|-
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 4 &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 四 &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; # &lt;/span&gt;
|| 2024 May 1st
|}
{|
|-
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 5 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 五 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; / &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 13px;&quot;&gt; ⓪ &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 12px;&quot;&gt; ① &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 11px;&quot;&gt; ② &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 10px;&quot;&gt; ③ &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 9px;&quot;&gt; ④ &lt;/span&gt;
||  &lt;span style=&quot;color:blue; font-size: 8px;&quot;&gt; ⓪ ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑬ ⑭ ⑮ ⑯ ⑰ ⑱ ⑲ ⑳ ㉑ ㉒ ㉓ ㉔ ㉕ ㉖ ㉗ ㉘ ㉙ ㉚ ㉛ ㉜ ㉝ ㉞ ㉟ ㊱ ㊲ ㊳ ㊴ ㊵ ㊶ ㊷ ㊸ ㊹ ㊺ ㊻ ㊼ ㊽ ㊾ ㊿ &lt;/span&gt;
|-
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 6 &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 六 &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; @ &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/ogberochan &lt;span style=&quot;color:#a020f0&quot;&gt; @ &lt;/span&gt;]
|style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/mrowrpurr &lt;span style=&quot;color:teal&quot;&gt; @ &lt;/span&gt;] 
|style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/spectraphonic @]
|style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/nazarenuuz @]
|style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/nazaretto333 @]
|style=&quot;background:#E6FFEA;&quot; | [https://www.twitch.tv/meleneth @]
|-
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 7 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 七 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; * &lt;/span&gt;
|| [https://susieshellenberger.com/?fbclid=IwZXh0bgNhZW0CMTAAAR2SR7v_qlePEJDASErzgJuoExGIfSO99g13OaQuJojyaO2VKYE5JXIZjaU_aem_AZmOheXKyqdRC5o-8dUIm1e-ZfdbQfHW8BQ7nGDRa_dDewMJq_N_xGFTO_SRj_4mBVx2T05euHX5hyEQW1wx37T8 pastor suizie]
|-
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 8 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 八 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt;  $ &lt;/span&gt;
| US$ || &lt;span style=&quot;color:red&quot;&gt; ¥ &lt;/span&gt;
|-
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 9 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 九 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; % &lt;/span&gt;
|-
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10 &lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 十 &lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; : &lt;/span&gt;
|-
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 零 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; _ &lt;/span&gt;
|-
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 百 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; ; &lt;/span&gt;
|style=&quot;background:#F2E0CE;&quot;| [https://www.latex-project.org/publications/ &lt;span style=&quot;color:teal&quot;&gt;LaTeX&lt;/span&gt;]
|-
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13  &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 千 &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ( ) &lt;/span&gt;
|| .pdf
|-
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14 &lt;/span&gt; 
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 羽 &lt;/span&gt; 
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; { } &lt;/span&gt; 
|| .png
|-
| style=&quot;background:indigo;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15 &lt;/span&gt;
| style=&quot;background:indigo;&quot; | &lt;span style=&quot;color:white&quot;&gt; 鶴 &lt;/span&gt;
| style=&quot;background:indigo;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ ] &lt;/span&gt;
|| ISBN
|}

&lt;hr&gt;
  @SalemFirstNaz https://linktr . ee/SalemFirstNaz?fbclid=IwZXh0bgNhZW0CMTAAAR3b0LPLXW-WwT8h8XJM2NsHvpjEFmLAw0hQwgxf2rYHgEA4az-2oa7q8wc_aem_AZlShbKjhUPc1b_IhKDDkGvNJQmSq6p1p4mJwHxtE0Cie1qFfHdqGTEbhpt65-AF0OV9bO7gMbGcrdqmYBBv58mn
&lt;hr&gt;


{|
|-
| style=&quot;background:indigo;&quot; | &lt;span style=&quot;color:white; font-size: 27px;&quot;&gt; 千羽ISBN鶴 &lt;/span&gt;
|| bibliography line
|-
|| 978-0-399-16657-0
|| Martin, Brian F. Invincible : The 10 Lies You Learn Growing up with Domestic Violence, and the Truths to Set You Free. New York, Perigee, 2015.
|-
|| 978-1-8380395-0-9
||
|}
&lt;hr&gt;

 publications https://www.ohnaz.com/?fbclid=IwZXh0bgNhZW0CMTAAAR1uqojWR1dTzqL8Qi8ZWsNnP9hCniv3PEfEnXohp3gufqxjSF0w0cCegXQ_aem_AWVg8huyBPKlH4pMBQlejepuIWscb_DllRHzgfv72Jt-lYWW8mQFH200UISkE_jd8jVpjQNwEZEVdbO4k9QcwBgY

 https://www.latex-project.org/publications/ -&gt; .pdf  -&gt; .png [https://pdfguru.com/app/pdf-to-png?id_partner=g_search&amp;feature=PDFToPNG&amp;clickid=Cj0KCQjwlZixBhCoARIsAIC745BHdGGj_Y_mlGeuxJu4zXuKtHXYTn72OXVlWkp4hwVjgqwVLCFyvrAaAgrCEALw_wcB&amp;campid=19964454624&amp;agid=152744234815&amp;keyword=pdf%20to%20png&amp;matchtype=e&amp;creatid=669641122537&amp;extid=&amp;targetid=kwd-302049935021&amp;device=c&amp;devmod=&amp;placement=&amp;adpos=&amp;gad_source=1&amp;gclid=Cj0KCQjwlZixBhCoARIsAIC745BHdGGj_Y_mlGeuxJu4zXuKtHXYTn72OXVlWkp4hwVjgqwVLCFyvrAaAgrCEALw_wcB png]
{|
|| 700 mm 
|| =
|| 27.5581 inches
|| = 
|| 27.5&quot; * 333dpi = 
|| by 
|| 1000 mm 
|| =
|| 39.3701 inches
|| =
|| 39.3&quot; * 333dpi = 
|-
||  
|}

{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 1.一&lt;/span&gt;
|| NewspaperClub || Traditional || mm || 380x578||760x1156 || 2028 || Leap Day || Broadsheet || INDESIGN TEMPLATE || SCRIBUS TEMPLATE || CANVA TEMPLATES
|-
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 2.二&lt;/span&gt;
|| NewspaperClub || Digital || mm || 350x500||700x1000|| 2024 || Almanac || Digital || INDESIGN TEMPLATE || SCRIBUS TEMPLATE || CANVA TEMPLATES
|-
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 3.三 &lt;/span&gt;
|| NewspaperClub || Traditional || mm || 380x578||760x1156|| 2024 || Summer || Broadsheet || INDESIGN TEMPLATE || SCRIBUS TEMPLATE || CANVA TEMPLATES
|-
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 4.四&lt;/span&gt;
|| Amazon || || 
|-
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 5.五 &lt;/span&gt;
|| Weekly || || || 00 || 11 || 22 || 33 || 44
|-
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 6.六&lt;/span&gt;
|-
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 7.七 &lt;/span&gt;
|-
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 8.八&lt;/span&gt;
|-
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 9.九 &lt;/span&gt;
|-
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10.十&lt;/span&gt;
|-
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11.零&lt;/span&gt;
|-
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12.百&lt;/span&gt;
|-
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13.千 &lt;/span&gt;
|-
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14.羽 &lt;/span&gt; 
|-
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15.鶴 &lt;/span&gt;
|}


{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 1.一&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 2.二&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 3.三 &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 4.四&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 5.五 &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 6.六&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 7.七 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 8.八&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 9.九 &lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10.十&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11.零&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12.百&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13.千 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14.羽 &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15.鶴 &lt;/span&gt;
|}
{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 16.月 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 17.火 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 18.水 &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 19.木 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 20.金 &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 21.土 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 22.日 ⽬ &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 23.者&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 24.振&lt;/span&gt; 
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 25.母&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 26.父&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 27.間&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 28.善 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 29.様&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 30.皆 &lt;/span&gt;
|}

{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 31.王&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 32.女&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 33.混&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 34.気&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 35.家&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 36.引&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 37.猫&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 38.沌&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 39.忍&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 40.伝&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 41.子&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 42.遺&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 43.人&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 44.仮&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 45.改 &lt;/span&gt;
|}
{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 46.黄&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 47.橙&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 48.赤&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 49.紫&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 50.紺&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 51.緑&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 52.白&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 53.黒&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 54.灰&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 55.桃 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 56.青&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 57.茶&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 58.銅&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 59.銀&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 60.色 &lt;/span&gt;
|}
{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 61.北&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 62.先&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 63.東&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 64.直&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 65.南&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 66.西&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 67.此 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 68.方&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 69.彼&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 70.年&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 71.週 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 72.何 &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 73.後&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 74.今&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 75.処 &lt;/span&gt;
|}
{|
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 76.上&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 77.毎&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 78.右&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 79.斜 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 80.下&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 81.左&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 82.朝&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 83.晩 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 84.去&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 85.昨&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 86.再&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 87.週&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 88.面&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 89.m̂&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 90.愛 &lt;/span&gt;
|}


== 2028==
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; 
|+ 2028 Leap Day l`chaim == 128.2.29
|-
| rowspan=&quot;101&quot; style=&quot;background:black; color: white&quot; |  לי חיים
| colspan=&quot;16&quot; style=&quot;background:black; color: white&quot; |  לי חיים
|-
| rowspan=&quot;10&quot; style=&quot;background:silver; color: indigo&quot; | indigo
| colspan=&quot;16&quot; style=&quot;background:silver; color: indigo&quot; |  silver
|-
|| K || Q || LL || X || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ( A )&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; { er } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ l.〇 ] &lt;/span&gt;
|-
| colspan=&quot;6&quot; style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 과로사 &lt;/span&gt;
| colspan=&quot;4&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 過勞死 &lt;/span&gt;
| colspan=&quot;3&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 勞 &gt; 劳 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:silver&quot; | &lt;span style=&quot;color:pink&quot;&gt; 〇 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:black&quot; | &lt;span style=&quot;color:white&quot;&gt; :⊙.⊙; &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://materializecss.com/icons.html 🟨]&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟧]&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://fonts.google.com/specimen/Cherry+Bomb+One 🟥]&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟪]&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟦]&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟩]&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://wikiconference.org/wiki/2023/Program/Submissions#top 🧿]&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; [https://calendar.google.com/calendar/u/0/r ⊙]&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikiversity.org/w/index.php?title=User:VeronicaJeanAnderson 人]&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10.十&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11.零&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12.百&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13.千 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14.羽 &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15.鶴 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 16.月 👀 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 17.火 👁 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 18.水 🥽  &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 19.木 👁‍🗨&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 20.金 👁️ &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 21.土 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 22.日 ⽬ &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 23.者&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 24.振&lt;/span&gt; 
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 25.母&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 26.父&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 27.間&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 28.善 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 29.様&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 30.皆 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 31.王&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 32.女&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 33.混&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 34.気&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 35.家&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 36.引&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 37.猫&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 38.沌&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 39.忍&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 40.伝&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 41.子&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 42.遺&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 43.人&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 44.仮&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 45.改 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 46.黄&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 47.橙&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 48.赤&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 49.紫&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 50.紺&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 51.緑&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 52.白&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 53.黒&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 54.灰&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 55.桃 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 56.青&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 57.茶&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 58.銅&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 59.銀&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 60.色 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 61.北&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 62.先&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 63.東&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 64.直&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 65.南&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 66.西&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 67.此 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 68.方&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 69.彼&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 70.年&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 71.週 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 72.何 &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 73.後&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 74.今&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 75.処 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 76.上&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 77.毎&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 78.右&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 79.斜 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 80.下&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 81.左&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 82.朝&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 83.晩 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 84.去&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 85.昨&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 86.再&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 87.週&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 88.面&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 89.m̂&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 90.愛 &lt;/span&gt;
|-
| colspan=&quot;12&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 93 ( )&lt;/span&gt;
| colspan=&quot;2&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 92 { } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 91 [ ] &lt;/span&gt;
|}


{|
|| a
|| a
|| a
|}

 un ob-fus-cat-ing  https://www.realtor.com/assumable https://jaso.org/ https://www.gamedeveloper.com/business/thor-pirate-software-leaving-aaa-to-go-indie https://www.ohnaz.com/?fbclid=IwZXh0bgNhZW0CMTAAAR1uqojWR1dTzqL8Qi8ZWsNnP9hCniv3PEfEnXohp3gufqxjSF0w0cCegXQ_aem_AWVg8huyBPKlH4pMBQlejepuIWscb_DllRHzgfv72Jt-lYWW8mQFH200UISkE_jd8jVpjQNwEZEVdbO4k9QcwBgY

 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC61387/

 https://meta.wikimedia.org/wiki/User:PurpleProjectsPDX97123

 https://en.wikipedia.org/wiki/Bridge_restaurant#/media/File:OHare_Oasis.jpg

 kansas

 #weAre [https://github.com/hahaveronicajin/faux-ing i am]

 Afrikaans | العربية | অসমীয়া | asturianu | azərbaycanca | Boarisch | беларуская | беларуская (тарашкевіца) | български | ပအိုဝ်ႏဘာႏသာႏ | বাংলা | བོད་ཡིག | bosanski | català | کوردی | corsu | čeština | Cymraeg | dansk | Deutsch | Deutsch (Sie-Form) | Zazaki | ދިވެހިބަސް | Ελληνικά | emiliàn e rumagnòl | English | Esperanto | español | eesti | euskara | فارسی | suomi | français | Nordfriisk | Frysk | galego | Alemannisch | ગુજરાતી | עברית | हिन्दी | Fiji Hindi | hrvatski | magyar | հայերեն | interlingua | Bahasa Indonesia | Ido | íslenska | italiano | 日本語 | ქართული | ភាសាខ្មែរ | 한국어 | Qaraqalpaqsha | kar | kurdî | Limburgs | ລາວ | lietuvių | Minangkabau | македонски | മലയാളം | молдовеняскэ | Bahasa Melayu | မြန်မာဘာသာ | مازِرونی | Napulitano | नेपाली | Nederlands | norsk nynorsk | norsk | occitan | Kapampangan | Norfuk / Pitkern | polski | português | português do Brasil | پښتو | Runa Simi | română | русский | संस्कृतम् | sicilianu | سنڌي | Taclḥit | සිංහල | slovenčina | slovenščina | Soomaaliga | shqip | српски / srpski | svenska | ꠍꠤꠟꠐꠤ | ślůnski | தமிழ் | тоҷикӣ | ไทย | Türkmençe | Tagalog | Türkçe | татарча / tatarça | ⵜⴰⵎⴰⵣⵉⵖⵜ  | українська | اردو | oʻzbekcha / ўзбекча | vèneto | Tiếng Việt | 吴语 | 粵語 | 中文（简体） | 中文（繁體） | +/-


=WeAre=
== ==
== ==
== ==
== ==
== ==
== ==

== i am ==
 https://github.com/hahaveronicajin/faux-ing
== Anderson ==
I was born an Anderson before the internet back in the days when bubbles rarely got popped.

Grandpa was an Anderson and made sure I took pride in the name whilst teasing that I wouldn't get to keep the name. I've managed to keep the name...

What's in a Name? &quot;Would a Rose smell as sweet?&quot; I think a name is useful identifying what we'd call &quot;classes&quot; of people if programming.

This season of my life, I've been watching for familiar names and taking pleasure in learning about new people because there are so many more Andersons that I'm not related to than that I am related to.

In Minnesota Karl Anderson is the prosecutor who took on the death of Isaac Michael Schuman and fellow river rafters on July 30th, 2022 = all 17 to 24. WCCO.com seems to be tracking the story and I've seen sparse bits of testimony along with Miu's initial interview which clearly shows he did not properly comprehend what had happened.

Nicolae Miu isn't the first criminal to have caught my eye over the years, I think time we spend &quot;being&quot; often coincides with time we are trying to fill with details. With all that time pumping milk with triplets in the NICU, I found myself learning about Andrea Yates. Wrapping my compassion in words that don't disrespect the obvious &quot;victims&quot; in cases like this has nothing to do with disrespecting those victims... Perhaps I'm overly pragmatic, but what I learned studying Hitler was that his problem wasn't with Jews - it was in the quantifiable (in cost of sick days and vodka) cost mistreating life cost his militia.

=== Isaac ===

What's in the Isaac? 

Abraham was challenged by God to kill his son and the story reads a ram is sacrificed instead. It's one of those names like Daemon, when I just always wondered a tad bit about people who chose such a name for their child, but Isaac Shuman's parents didn't expect for a stranger to take the life of their son.

== https://jaso.org/ ==
https://jaso.org/

  Japan-America Society of Oregon | 900 SW Fifth Avenue, Suite #1810 | Portland, Oregon 97204 Phone: 503.552.8811
© 2024 Japan-America Society of Oregon

== https://home.unicode.org/ ==
https://home.unicode.org/

== ==
I see you reverted a few things on my user page. I wanted to assure you that I don't think anyone has edited my page other than me and now, you.  Frankly, I'm pleased to interact with you! I have been a Trump Hostage from 2017 until March 7th 2024. NW Forensic's Dr Jennifer Johnson can tell you who I am and what my position is in a legal battle for free speech in Oregon. I couldn't work my wikiversity project until I was no longer under threat. It seems that I am finally no longer under threat. 

Clarification: I was a Trump hostage because of my opposition to Trump and Putin and Hamas. The War Crimes I witnessed have been reported to the appropriate agencies internationally.

I am bringing together Oregon, USA 97305 and Chiba-ken, Japan 272 using Rotarians, Nazarenes, and academics. We are now working on a almanac that will be put into the creative commons for print wherever it's wanted focusing on ways we can interact with each other to encourage social security.

https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson#WeAre &lt;-- this is the link I just put in a Facebook chat where I am talking to real people about how to use our coming funds in a public way - hopefully, creating content here in this wiki-verse - after all, I think wiki originated here in Portland, OR.

I did not expect to get stonewalled from 2017 until 3/7/2024. For my ruminations while under threat, I've tried to keep them in my user space per a previous custodian. 

I'm working on the technical relationship between wiki and LaTeX and printing 1000s of copies of our 1st annual almanac now. We intend to be print ready by the end of July to be delivered to Nazarenes and Rotarians over the winter using paper that is easy and safe to burn.

I don't know if this is the right place to put this message - if you would like to email me I'm at oregongyre.berochan@gmail.com; and I will be streaming now that I'm no longer a Trump Hostage here: https://www.twitch.tv/ogberochan after I get some hardware moved.

Thank you for your efforts!
dozo yoroshiku onegaishimasu
[[User:PurpleProjectsPDX97123|PurpleProjectsPDX97123]] ([[User talk:PurpleProjectsPDX97123|discuss]] • [[Special:Contributions/PurpleProjectsPDX97123|contribs]]) 21:22, 20 March 2024 (UTC)

== Guns in Discord ==
=== March 0 ===
=== March 1 ===
=== March 2 ===
=== March 3 ===
=== March 0100 ===
=== March 100 ===

=== March 10x ===
=== March 11x ===
=== March 120 ===
=== March 121 ===
=== March 122 ===


=== March 123 ===

=== March 124 ===
 3/31 basicBeekeeper

 a m an a p lanac a nal pa nam a .json

 Loren Anderson Obituary
 Loren Charles Anderson September 12, 1927 - August 28, 2010 SALEM - Loren Anderson, age 82, beloved husband, father, grandfather, and great-grandfather, died at Salem hospital on Saturday, August 28, 2010, with family by his side. Loren was a member of Salem First Church of the Nazarene where he served faithfully over his lifetime. Loren served in the Merchant Marines during WWII on the ship SS Samuel Gompers where he survived a torpedo attack from a Japanese submarine. Loren then pastored churches in Gibson City, Moweaque, and Stonington, Ill and then followed his passion to teach by becoming a science teacher at Glenbard East High School in Lombard, Ill. Loren obtained his doctorate of Science Education and then was nicknamed &quot;Doc Anderson&quot;. Loren retired to Salem, Oregon and continued teaching at Christian Center and Salem Academy. His teaching career totaled 40 years. Loren joins his beloved wife of 62 years, Juanita Anderson, who passed away on February 23, 2010. Loren is survived by his children, Cheryl Anderson, Charles and Bonita Anderson, Byron and Bonnie Anderson, and Valerie and Paul McCallum; and daughter-in-law, Cynthia Anderson. He is further survived by 21 grandchildren, 19 great-grandchildren and many friends. A graveside service will be held at 11 a.m., Tuesday, August 31, 2010, at Belcrest Memorial Park, 1295 Browning Avenue South, Salem, Oregon. Friends and family are welcome. Donations can be made in his honor to Salem First Church of the Nazarene. Arrangements by Howell-Edwards-Doerksen with Rigdon-Ransom Funeral Directors.
 Published by The Statesman Journal on Aug. 31, 2010.

 https://www.legacy.com/us/obituaries/statesmanjournal/name/loren-anderson-obituary?id=27541895&amp;_gl=1*19vxs9m*_gcl_au*MTM3NjQzMjI4NS4xNzExNzQyNTE2
 

 3/30 Sequoia
 3/29 &quot;Voting is Violence&quot;
 3/28  yes

== Oregon Gyre Almanac ==
 https://www.newspaperclub.com/choose/broadsheet/traditional
 https://gdkp4t1dnadhd.blogspot.com/?fbclid=IwAR1BdqyyT7YuilmUTvq3vbIdFyVy4gGvTS5OlRa3plPmkvss51Co_9MmWpY_aem_AcvhmythGHUGVp3jFDnP63jNFPVJD3KRO0danTtlR8OIItlZl3XX9NTl8Il3RPHlugVXwzLL4X88bOpES4935nTy

=== Printer === 
 https://www.newspaperclub.com
 size: Traditional (1000 copies)
 paper: recycled
 size: 
 pages: multiples of 4; probably 12.
 dpi: 300
 smallest font: 8pt
 centrespread: yes
 margins 15mm

= 2028 Leap Day l`chaim == 128.2.29 Table =
== 2028 Leap Day l`chaim == 128.2.29 Table ==
=== 2028 Leap Day l`chaim == 128.2.29 Table ===
====2028 Leap Day l`chaim == 128.2.29 Table ====
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; 
|+ 2028 Leap Day l`chaim == 128.2.29
|-
| rowspan=&quot;101&quot; style=&quot;background:black; color: white&quot; |  לי חיים
| colspan=&quot;16&quot; style=&quot;background:black; color: white&quot; |  לי חיים
|-
| rowspan=&quot;10&quot; style=&quot;background:silver; color: indigo&quot; | indigo
| colspan=&quot;16&quot; style=&quot;background:silver; color: indigo&quot; |  silver
|-
|| K || Q || LL || X || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ( A )&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; { er } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ l.〇 ] &lt;/span&gt;
|-
| colspan=&quot;6&quot; style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 과로사 &lt;/span&gt;
| colspan=&quot;4&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 過勞死 &lt;/span&gt;
| colspan=&quot;3&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 勞 &gt; 劳 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:silver&quot; | &lt;span style=&quot;color:pink&quot;&gt; 〇 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:black&quot; | &lt;span style=&quot;color:white&quot;&gt; :⊙.⊙; &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://materializecss.com/icons.html 🟨]&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟧]&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://fonts.google.com/specimen/Cherry+Bomb+One 🟥]&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟪]&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟦]&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟩]&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://wikiconference.org/wiki/2023/Program/Submissions#top 🧿]&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; [https://calendar.google.com/calendar/u/0/r ⊙]&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikiversity.org/w/index.php?title=User:VeronicaJeanAnderson 人]&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10.十&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11.零&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12.百&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13.千 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14.羽 &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15.鶴 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 16.月 👀 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 17.火 👁 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 18.水 🥽  &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 19.木 👁‍🗨&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 20.金 👁️ &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 21.土 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 22.日 ⽬ &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 23.者&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 24.振&lt;/span&gt; 
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 25.母&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 26.父&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 27.間&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 28.善 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 29.様&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 30.皆 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 31.王&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 32.女&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 33.混&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 34.気&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 35.家&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 36.引&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 37.猫&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 38.沌&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 39.忍&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 40.伝&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 41.子&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 42.遺&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 43.人&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 44.仮&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 45.改 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 46.黄&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 47.橙&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 48.赤&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 49.紫&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 50.紺&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 51.緑&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 52.白&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 53.黒&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 54.灰&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 55.桃 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 56.青&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 57.茶&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 58.銅&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 59.銀&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 60.色 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 61.北&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 62.先&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 63.東&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 64.直&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 65.南&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 66.西&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 67.此 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 68.方&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 69.彼&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 70.年&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 71.週 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 72.何 &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 73.後&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 74.今&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 75.処 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 76.上&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 77.毎&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 78.右&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 79.斜 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 80.下&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 81.左&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 82.朝&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 83.晩 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 84.去&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 85.昨&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 86.再&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 87.週&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 88.面&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 89.m̂&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 90.愛 &lt;/span&gt;
|-
| colspan=&quot;12&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 93 ( )&lt;/span&gt;
| colspan=&quot;2&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 92 { } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 91 [ ] &lt;/span&gt;
|}

= 2024 Leap Day l`chaim == 124.2.29 Table =
== 2024 Leap Day l`chaim == 124.2.29 Table ==
=== 2024 Leap Day l`chaim == 124.2.29 Table ===
====2024 Leap Day l`chaim == 124.2.29 Table ====
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; 
|+ 2024 Leap Day l`chaim == 124.2.29
|-
| rowspan=&quot;101&quot; style=&quot;background:black; color: white&quot; |  לי חיים
| colspan=&quot;16&quot; style=&quot;background:black; color: white&quot; |  לי חיים
|-
| rowspan=&quot;10&quot; style=&quot;background:silver; color: indigo&quot; | indigo
| colspan=&quot;16&quot; style=&quot;background:silver; color: indigo&quot; |  silver
|-
|| K || Q || LL || X || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ( A )&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; { er } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ l.〇 ] &lt;/span&gt;
|-
| colspan=&quot;6&quot; style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 과로사 &lt;/span&gt;
| colspan=&quot;4&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 過勞死 &lt;/span&gt;
| colspan=&quot;3&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 勞 &gt; 劳 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:silver&quot; | &lt;span style=&quot;color:pink&quot;&gt; 〇 &lt;/span&gt; 
| colspan=&quot;1&quot; style=&quot;background:black&quot; | &lt;span style=&quot;color:white&quot;&gt; :⊙.⊙; &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://materializecss.com/icons.html 🟨]&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟧]&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://fonts.google.com/specimen/Cherry+Bomb+One 🟥]&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟪]&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟦]&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikipedia.org/wiki/Circled_dot 🟩]&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://wikiconference.org/wiki/2023/Program/Submissions#top 🧿]&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; [https://calendar.google.com/calendar/u/0/r ⊙]&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; [https://en.wikiversity.org/w/index.php?title=User:VeronicaJeanAnderson 人]&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 10.十&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 11.零&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 12.百&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 13.千 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 14.羽 &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 15.鶴 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 16.月 👀 &lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 17.火 👁 &lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 18.水 🥽  &lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 19.木 👁‍🗨&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 20.金 👁️ &lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 21.土 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 22.日 ⽬ &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 23.者&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 24.振&lt;/span&gt; 
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 25.母&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 26.父&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 27.間&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 28.善 &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 29.様&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 30.皆 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 31.王&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 32.女&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 33.混&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 34.気&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 35.家&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 36.引&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 37.猫&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 38.沌&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 39.忍&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 40.伝&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 41.子&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 42.遺&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 43.人&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 44.仮&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 45.改 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 46.黄&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 47.橙&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 48.赤&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 49.紫&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 50.紺&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 51.緑&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 52.白&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 53.黒&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 54.灰&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 55.桃 &lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 56.青&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 57.茶&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 58.銅&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 59.銀&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 60.色 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 61.北&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 62.先&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 63.東&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 64.直&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 65.南&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 66.西&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 67.此 &lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 68.方&lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 69.彼&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 70.年&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 71.週 &lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 72.何 &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 73.後&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 74.今&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 75.処 &lt;/span&gt;
|-
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 76.上&lt;/span&gt;
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 77.毎&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; 78.右&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 79.斜 &lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 80.下&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; 81.左&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 82.朝&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; 83.晩 &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; 84.去&lt;/span&gt;
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; 85.昨&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; 86.再&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; 87.週&lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 88.面&lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 89.m̂&lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 90.愛 &lt;/span&gt;
|-
| colspan=&quot;12&quot; style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; 93 ( )&lt;/span&gt;
| colspan=&quot;2&quot; style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; 92 { } &lt;/span&gt; 
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; 91 [ ] &lt;/span&gt;
|}


== mono koto ma and mu ==
 物 (mono) 事 (koto) 間 (ma) 無 (mu)

{|
|-
|| ||
|| 1
|| 2 
|| 3
|-
|| ||
|| 🟨 
|| 🟥 
|| 🟦
|-
|style=&quot;background:#000; color: silver; font-size: 111px;&quot;| 無
|style=&quot;background:#000; color: gold; font-size: 44px;&quot;| ★ ☆ ✪ 
|style=&quot;background:#80808080; color: khaki; font-size: 111px;&quot;| 物
|style=&quot;background:#80808080; color: coral; font-size: 111px;&quot;| 事
|style=&quot;background:#80808080; color: cornflowerblue; font-size: 111px;&quot;| 間


|-
|||||style=&quot;background:; color:#; font-size: 11px;&quot;| (mono)  
|style=&quot;background:; color:#; font-size: 11px;&quot;| (koto) 
|style=&quot;background:; color:#; font-size: 11px;&quot;| (ma)
|-
|||||style=&quot;background:; color:#; font-size: 7px;&quot;| tangible thing
|style=&quot;background:; color:#; font-size: 7px;&quot;| intangible thing
|style=&quot;background:; color:#; font-size: 7px;&quot;| everything in between
|-
|||||style=&quot;background:; color:#; font-size: 11px;&quot;| khaki
|style=&quot;background:; color:#; font-size: 11px;&quot;| coral
|style=&quot;background:; color:#; font-size: 11px;&quot;| cornflowerblue
|}

 @Unite4Copyright

 0⬤◯⓪
 Every whole circle includes a never ending irrational piece we call pi
 🥧 π 𝜋 𝛑 𝝅 𝞹 ℼ

 [https://www.w3schools.com/colors/default.asp CSS Color Values]
 With CSS, colors can be specified in different ways:
 
 By color names (140)
 As RGB values
 As hexadecimal values
 As HSL values (CSS3)
 As HWB values (CSS4)
 With the currentcolor keyword


{|
|-
|| 1
|| 2 
|| 3
|-
|| 🟨 
|| 🟥 
|| 🟦
|-
|style=&quot;background:#80808080; color: khaki; font-size: 111px;&quot;| 物
|style=&quot;background:#80808080; color: coral; font-size: 111px;&quot;| 事
|style=&quot;background:#80808080; color: cornflowerblue; font-size: 111px;&quot;| 間
|-
|style=&quot;background:; color:#; font-size: 11px;&quot;| (mono)  
|style=&quot;background:; color:#; font-size: 11px;&quot;| (koto) 
|style=&quot;background:; color:#; font-size: 11px;&quot;| (ma)
|-
|style=&quot;background:; color:#; font-size: 7px;&quot;| tangible thing
|style=&quot;background:; color:#; font-size: 7px;&quot;| intangible thing
|style=&quot;background:; color:#; font-size: 7px;&quot;| everything else that disappears
|-
|style=&quot;background:; color:#; font-size: 11px;&quot;| khaki
|style=&quot;background:; color:#; font-size: 11px;&quot;| coral
|style=&quot;background:; color:#; font-size: 11px;&quot;| cornflowerblue
|}

=1=
{|
|-
|| 1
|| 2 
|| 3
|-
|| 🟨 
|| 🟥 
|| 🟦
|-
|style=&quot;background:; color:#; font-size: 33px;&quot;| 物
|style=&quot;background:; color:#; font-size: 33px;&quot;| 事
|style=&quot;background:; color:#; font-size: 33px;&quot;| 間
|-
|style=&quot;background:; color:#; font-size: 11px;&quot;| (mono)  
|style=&quot;background:; color:#; font-size: 11px;&quot;| (koto) 
|style=&quot;background:; color:#; font-size: 11px;&quot;| (ma)
|}

=2=
{|
|-
|| 1
|| 2 
|| 3
|-
|| 🟨 
|| 🟥 
|| 🟦
|-
|| 4
|| 5
|| 6
|-
|| 🟧
|| 🟪
|| 🟩
|}
=3=
 /əˈpäləjəst/
 a·pol·o·gist
 noun
 a person who offers an argument in defense of something controversial.
 &quot;critics said he was an apologist for colonialism&quot;
 
 defender supporter upholder advocate proponent exponent
{|
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|| 2 
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|-
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟦
|| 🟩
|}
=for=
 https://www.unicode.org/charts/beta/nameslist/n_1D100.html
{|
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|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|}

=5=
{|
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|-
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|}
{|
|-
|| [https://www.youtube.com/@andymation 1]
|| 2
|| 3
|| 4
|| 5
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|| 7
|| 8
|| 9
|| 10
|| 11
|| 12
|| 13
|| 14
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|-
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|| 🟨
|| 🟧
|| 🟥
|| 🟪
|}

=📅=
==📆 🗓==

==124==
=== 124 == 2024 Gregorian calendar ===
 2024 Gregorian calendar
===月===
 12 months
{|
|-
|| ㋀
|| ㋁
|| ㋂
|| ㋃
|| ㋄
|| ㋅
|| ㋆
|| ㋇
|| ㋈
|| ㋉
|| ㋊
|| ㋋
|}

===週===
 52 weeks

=kana=
==katakana==
 ゠	
 ァ	ア	ィ	イ	ゥ	ウ	ェ	エ	ォ	オ	
 カ	ガ	キ	ギ	ク グ ケ	ゲ	コ	ゴ	
 サ	ザ	シ	ジ	ス	ズ	セ	ゼ	ソ	ゾ
 タ ダ	チ	ヂ	ッ	ツ	ヅ	テ	デ	ト	ド
 ナ	ニ	ヌ	ネ	ノ
 ハ バ	パ	ヒ	ビ	ピ	フ	ブ	プ	ヘ	ベ	ペ	ホ	ボ	ポ
 マ	ミ ム	メ	モ
 ャ	ヤ	ュ	ユ	ョ	ヨ
 ラ	リ	ル	レ	ロ	ヮ	ワ
 ヰ	ヱ	ヲ	ン	ヴ	ヵ	ヶ	ヷ	ヸ	ヹ	ヺ	・	ー	ヽ	ヾ	ヿ

== ℤ ℕ ℙ ℚ ℝ ℂ ℍ 𝕊 ℑ ℜ ⅅ ⅆ ⅇ ⅈ ⅉ ⅈ ℵ ℭ==

 ℤ integers.
 ℕ natural numbers.
 ℙ primes.
 ℚ be rational.
 ℝ ℂ ℍ 𝕊 ℑ ℜ ⅅ ⅆ ⅇ ⅈ ⅉ ⅈ ℵ ℭ


 ℤ
 integers.
 ℕ
 natural numbers.
 ℙ
 primes.
 ℚ
 be rational.
 ℝ
 ℂ
 ℍ
 𝕊 
 ℑ
 ℜ
 ⅅ
 ⅆ
 ⅇ
 ⅈ
 ⅉ
 ⅈ
 ℵ
 ℭ


 Meaning in Math
 ℤ
 integers.
 ℕ
 natural numbers.
 ℙ
 primes.
 ℚ
 be rational.
 ℝ
 get real.
 ℂ
 complex number.
 ℍ
 quaternions.
 𝕊 
 sedenions.
 ℑ
 imaginary part
 ℜ
 real part
 ⅅ
 Derivative
 ⅆ
 Differential
 ⅇ
 euler's number (natural growth number)
 ⅈ
 imaginary unit.
 ⅉ
 notation used by engineers for ⅈ
 ℵ
 cardinality of infinite sets.
 ℭ
 continuum 

 [http://xahlee.info/comp/unicode_math_font.html source]

==📅==

=binary=
{|
|-
|style=&quot;background:; color:#; font-size: 22px;&quot;| ⬤
|| |||| |||| |||| |||| || || 
|-
|style=&quot;background:; color:#; font-size: 22px;&quot;| ◯
|| |||| |||| |||| |||| || || 
|-
|style=&quot;background:; color:#; font-size: 22px;&quot;| ⓪
|| |||| |||| |||| |||| || || 
|-
|style=&quot;background:indigo; color:#fff; font-size: 22px;&quot;| 一
|| ♡
|| ♥
|| NNU
|| 623 S University Blvd
|| Nampa
|| ID 
|| 83686
|| USA
|| GPS
|| Lat Long
|| geocache
|}
{|
|-
|| 二
||♤ 
||♠ 
|| NICH
|| @OHSU
|-
||||||||
♢ 三 ♦ LifeWorksNW
|-
||||||||
♧ 四 ♣ PSU
|-
||||||||
☯ 五 ☯ 1550
|-
||||||||
☮ 六 ☮ 1221
|-
||||||||
⛯ 七 ⛯Naz HQ Market St
|-
||||||||
♾ 八 ♾ MAPS 
|-
||||||||
⛾ 九 ⛾Nutrition First
|-
||||||||
⚜ 十 ⚜ Synagogue
|-
||||||||
⛩ 零 ⛩ Japanese Garden
|-
||||||||
⚽︎ 百 ⚽ Gilbert House
|-
||||||||
⛫ 千 ⛫ OHSU
|-
||||||||
⛼ 羽 ⛼ Copper Tins @ Oregon State Hospital
|-
||||||||
⛬ 鶴 ⛬ Hiroshima
|-
|| ☾ 月 ☽
|| ☾ 月 ☽
|| ☾ 月 ☽
|| 5150
|| % Mahalo 

|-
||♨ 火 ♨
|-
||||||||

⛆ 水 ⛆
|-
||

Ѫ 木 Ѫ
|-
||

⛀ 金 ⛂
|-
||

⛰ 土 ⛰
|-
||

☼ 日 ☀
|-
||

⚐ 者 ⚑
|-
||

⚞ 振 ⚟
|-
||

☁ 母 ☁
|-
||

⚾︎ 父 ⚾︎ 
|-
||

⛋ 間 ⛋
|-
||

善 
|-
||

様 
|-
||

皆 
|-
||

王 
|-
||

女 
|-
||

混 
|-
||

気 
|-
||

家 
|-
||

引 
|-
||

猫 
|-
||

沌 
|-
||

忍 
|-
||

伝 
|-
||

子
|-
||

|-
||
遺 
|-
||

人 
|-
||

仮 
|-
||

改 
|-
||

黄 
|-
||

橙 
|-
||

赤 
|-
||

紫 
|-
||

紺 
|-
||

緑 
|-
||

白 
|-
||

黒 
|-
||

灰 
|-
||

桃 
|-
||

青 
|-
||

茶 
|-
||

銅 
|-
||

銀 

色 

北 

先 

東 

直 

南 

西 

此 

方 

彼 

年 

週 

何 

後 

今 

処 

上 

毎 

右 

斜 

下

左 

朝 

晩 

去 

昨 

再 

週 

面 

m̂ 
|-
|||||||||||| 愛


|}

==📆==
==🗓==

 [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/old old]
 [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/simplify simplify]
=z=
 The Office of The Acting Duke and Duchess of Oregon
[https://en.wikiversity.org/wiki/User:OregonGyre#AC124_Almanac_for_AC125 AC124 Almanac for AC125]
 0;123-4_n

 https://en.wikiversity.org/wiki/User:OregonGyre

 人 : I am an individual. OregonGyre is a collaborative project. I'm working towards enticing 

 [hitler's favorite artist] Caspar David Friedrich died in poverty in 1840

background: hsl(60deg-270deg 50% 25-75%);
{|
|-
|style=&quot;color: hsl(60deg 50% 25%); font-size: 55px;&quot;|𝅜
|style=&quot;color: hsl(90deg 50% 25%); font-size: 55px;&quot;|𝅝
|style=&quot;color: hsl(120deg 50% 25%); font-size: 55px;&quot;|𝅗𝅥	
|style=&quot;color: hsl(150deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥	
|style=&quot;color: hsl(180deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥𝅮	
|style=&quot;color: hsl(210deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥𝅯	
|style=&quot;color: hsl(230deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥𝅰
|style=&quot;color: hsl(250deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥𝅱
|style=&quot;color: hsl(270deg 50% 25%); font-size: 55px;&quot;|𝅘𝅥𝅲
|-
|style=&quot;color: hsl(60deg 50% 40%); font-size: 55px;&quot;|𝅜
|style=&quot;color: hsl(90deg 50% 40%); font-size: 55px;&quot;|𝅝
|style=&quot;color: hsl(120deg 50% 40%); font-size: 55px;&quot;|𝅗𝅥	
|style=&quot;color: hsl(150deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥	
|style=&quot;color: hsl(180deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥𝅮	
|style=&quot;color: hsl(210deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥𝅯	
|style=&quot;color: hsl(230deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥𝅰
|style=&quot;color: hsl(250deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥𝅱
|style=&quot;color: hsl(270deg 50% 40%); font-size: 55px;&quot;|𝅘𝅥𝅲
|-
|style=&quot;color: hsl(60deg 50% 50%); font-size: 55px;&quot;|𝅜
|style=&quot;color: hsl(90deg 50% 50%); font-size: 55px;&quot;|𝅝
|style=&quot;color: hsl(120deg 50% 50%); font-size: 55px;&quot;|𝅗𝅥	
|style=&quot;color: hsl(150deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥	
|style=&quot;color: hsl(180deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥𝅮	
|style=&quot;color: hsl(210deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥𝅯	
|style=&quot;color: hsl(230deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥𝅰
|style=&quot;color: hsl(250deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥𝅱
|style=&quot;color: hsl(270deg 50% 50%); font-size: 55px;&quot;|𝅘𝅥𝅲
|-
|style=&quot;color: hsl(60deg 50% 60%); font-size: 55px;&quot;|𝅜
|style=&quot;color: hsl(90deg 50% 60%); font-size: 55px;&quot;|𝅝
|style=&quot;color: hsl(120deg 50% 60%); font-size: 55px;&quot;|𝅗𝅥	
|style=&quot;color: hsl(150deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥	
|style=&quot;color: hsl(180deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥𝅮	
|style=&quot;color: hsl(210deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥𝅯	
|style=&quot;color: hsl(230deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥𝅰
|style=&quot;color: hsl(250deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥𝅱
|style=&quot;color: hsl(270deg 50% 60%); font-size: 55px;&quot;|𝅘𝅥𝅲
|-
|style=&quot;color: hsl(60deg 50% 75%); font-size: 55px;&quot;|𝅜
|style=&quot;color: hsl(90deg 50% 75%); font-size: 55px;&quot;|𝅝
|style=&quot;color: hsl(120deg 50% 75%); font-size: 55px;&quot;|𝅗𝅥	
|style=&quot;color: hsl(150deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥	
|style=&quot;color: hsl(180deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥𝅮	
|style=&quot;color: hsl(210deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥𝅯	
|style=&quot;color: hsl(230deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥𝅰
|style=&quot;color: hsl(250deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥𝅱
|style=&quot;color: hsl(270deg 50% 75%); font-size: 55px;&quot;|𝅘𝅥𝅲
|-
|| 2/1
|| 1/1
|| 1/2
|| 1/4
|| 1/8
|| 1/16
|| 1/32
|| 1/64
|| 1/128
|}</text>
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== Licensing ==
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{{Information
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|Source={{own}}
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|Permission=public domain
}}
== Licensing ==
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      <text bytes="46189" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Dan Polansky!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOUI JS signature icon LTR.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
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&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:12, 17 March 2020 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
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== COVID-19 ==

Thank you Dan for the support and contribution to the [[COVID-19]] learning resources. I appreciate your contributions very much. Would like to coordinate collaborative effort a bit due to the dynamic change of COVID-19 situation globally. Is there a specific subpage (not user-page) that is content driven, that you would like to add you expertise e.g. Data Analysis?. Shall we revise the structure to be more user-friendly for finding specific learning resources? Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 13:10, 17 March 2020 (UTC)
: Thank you. As for Wikiversity, I really do not know how things are working here. I have encouraged another editor to create [[COVID-19/Julian Mendez]], which is original research and is super interesting. However, I have not reviewed the material much, just had a superficial glance. On the surface, the thinking is good: it emphasizes time lag of detection of a rapidly exponentially increasing phenomenon.
: Since Wikiversity allows original research, it presents a unique opportunity for material like  [[COVID-19/Julian Mendez]]. My experience from data analysis is somewhat limited; I have a pretty strong mathematical background, so I know that the derivative of ''a^x'' is ''ln a * a^x'', and that gives a super scary light on the covid thing, how both totals and daily increases have the same base of exponential growth until mitigated, whether cases or deaths.
: I probably do not have the energy to coordinate efforts and I have no improvement proposals on [[COVID-19]] structure and content. I am wondering when I am finally going to run out of gumption. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 17 March 2020 (UTC)

:: Thank you, no worries. Exponential growth might not be the appropriate mathematical model to describe the development of COVID-19 or an epidemiological outbreak in general, because developement of count has limits of growth e.g. the total population on earth. So logistical growth with a capacity is more likely to describe the development. Currently in the early phase the data shows an exponential pattern but closer to the capacity the derivation gets smaller and closer to zero. The question is, what is the capacity of the logistical growth. Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:24, 17 March 2020 (UTC)
::: The initial phase is exponential, and this is easily empirically verified by observing the straight lines in the graphs with logarithmic y-axis, but it is true that once factors limiting the growth set in, it ceases to be exponential. Without intervention/mitigation, starting to run out of people to infect is the main limiting factor of the exponential growth, from what I can see. However, this is where we do not want to get since that becomes numerically significant only after, say, 10% of the population gets infected, and luckily enough, hardly any country has come close to that degree. And this would be an interesting mathematical/epidemiological assignment for a classroom: determine at which degree of population penetration an unmitigated infection growth ceases significantly to be exponential, for some value of &quot;significantly&quot;. That would need to assume some model of spread; I was thinking of molecules in a gas hitting one another, but the social phenomenon may look much different because of non-Gaussian distribution of &quot;influencers&quot;, as it were; there would be some people who have hugely many contacts and targetting them specifically for isolation could make huge difference, and you do not get that in a gas, I suppose. But the gas model need not be so bad to get a very first idea, and isolating &quot;influencers&quot; would not need to suffice at all. I don't really know, but I do maintain that the virus growth is in an exponential phase and, unless mitigated, would stay in the exponential phase in the coming weeks in most countries that would not do mitigation. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:40, 17 March 2020 (UTC)
: For reference, I keep on expanding the following pages: [[COVID-19/Dan Polansky]] and [[Talk:COVID-19/Dan Polansky]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:39, 3 May 2020 (UTC)
: ''Late retraction'': As for &quot;The initial phase is exponential, and this is easily empirically verified by observing the straight lines in the graphs with logarithmic y-axis, but it is true that once factors limiting the growth set in, it ceases to be exponential&quot;: that is wrong or misleading; while the confirmed cases were indeed originally growing exponentially, the observed rate of exponential growth was due to exponential growth in number of tests, as is confirmed by observing test positivity rate and observing the rate of growth of tests. The true infection count could either be initially growing exponentially at hugely slower rate than the nominal confirmed cases, or they were growing according to Gompertz curve (see research by Michael Levitt) and therefore never growing exponentially. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:28, 15 August 2020 (UTC)

=== Effiency of Lock Down ===
thank you for adding that important topic, to COVID-19 learning resource, Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:13, 23 July 2020 (UTC)

== Transgenderism ==

I am experimenting with the following page: [[User:Dan Polansky/Transgenderism]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:48, 17 July 2020 (UTC)

== COVID-19 Data ==

Is the COVID-19 data you've added available in Wikidata? It would be better to have the data there and query it rather than saved as pages here. As Wikidata, anyone could query it in any language and for any Wikimedia project. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:33, 14 August 2020 (UTC)
: Not that I know of. I see your point with cross-wiki query and avoidance of duplication of storage. On the other hand, storing comma-separated lists of values directly in the wiki markup is very simple and convenient, and one can very easily take that and calculate e.g. moving averages from that. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:13, 15 August 2020 (UTC)

::This gets a little tricky, because [[Wikiversity:What Wikiversity is not|Wikiversity does not]] duplicate other Wikimedia projects. I haven't done a large data import and query like this, but I'm willing to try one and see if I can produce the same results using the preferred structure. Is there a particular page and data set you would recommend as a starting point? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:47, 15 August 2020 (UTC)
::: You may try [[COVID-19/All-cause deaths/London]] if you wish; it uses sources different from the other pages. Let me note that I am not very enthusiastic; I am afraid of making simple things more complex at just a little benefit. If, say, I will want to update the London data as I did today, instead of doing something utterly simple and straightforward I will either need to ask you for help or learn about Wikidata imports myself. The loss of personal productivity is likely to be non-trivial and may be a showstoper for me; instead of updating London, I would then do something more straightfoward. And when I learn how to do Wikidata and someone else wants to update London, it is now them who has to learn Wikidata.
::: What would be really useful would be to update the charting add-in in Wikiversity to be on par with Wikipedia: in Wikipedia, it produces raster images whereas in Wikiversity, it produces some kind of semi-live object that seems to take longer to load. (I misspoke; the thing produced in Wikiversity is a PNG image as well; it does not have antialised fonts in the x-axis when the labels are at angle. Wikipedia seems more up to date with the add-in.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:03, 15 August 2020 (UTC)

::::I've updated [[Module:Graph]]. If that's not it, you'll need to be more specific as to what is more current at Wikipedia so it can be imported. You can also request imports yourself at [[Wikiversity:Import]]. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:17, 15 August 2020 (UTC)
::::: Thank you! It did not help: the x-axis fonts are still not antialiased and there is still a change to red color on mouseover over the blue line. [[Module:Graph]] is only a layer over the extension itself ([https://www.mediawiki.org/wiki/Extension:Graph Extension:Graph]), and maybe the extension needs an update. Theoretically, [[Template:Graph:Chart]] might need an update as well. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:28, 16 August 2020 (UTC)

::::::See [[Special:Version]]. Extension:Graph appears to be current. I checked the dates on Template:Graph:Chart initially and it was also current. I'm happy to import whatever we need to update, but I'll need your help to find whatever that might be. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:43, 16 August 2020 (UTC)
: (Outdent) Let me double check:
:* Per [[Special:Version]], Extension:Graph: Wikiversity: (9e762ac) 06:27, 6 August 2020; Wikipedia: (9e762ac) 06:27, 6 August 2020
:* [[Module:Graph]], textual comparison between WV and WP: same
:* [[Template:Graph:Chart]], textual comparison between WV and WP: same
:* [[Template:Graph:Chart/styles.css]], textual comparison between WV and WP: same except for a comment line, immaterial
:* Raw test of the JSON markup in [[User:Dan Polansky/sandbox]]: no x-axis antialiasing
: Could there be a relevant setting at LocalSettings.php? [https://www.mediawiki.org/wiki/Extension_talk:Graph Mediawiki's Extension talk:Graph] mentions $wgGraphImgServiceUrl, so something like that. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:34, 17 August 2020 (UTC)

::At this point, the best option would be to file a [[phabricator:]] ticket and see if one of the developers can identify the problem. We can't make php setting changes, so the ticket will be necessary anyway. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:26, 17 August 2020 (UTC)
::: It seems that on Wikiversity, the charts are made using ''canvas'' element, and they are plotted by the client browser via Javascript. By contrast, the English Wikipedia seems to be customized to have a server backend generate the PNG images for the charts so the client browser does not have to do any plotting, showin an ''img'' element instead. One consequence is that Wikipedia charts show fine on older devices whose browsers do not support canvas element. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:14, 8 September 2020 (UTC)

== Original research ==

For my reference:
* [[Wikiversity:Original research]]
*: &quot;Original research which meets the guidelines of this policy is permitted on Wikiversity. Researchers devoted to scholarly investigation using sound, ethical methods are encouraged to develop and disseminate their work via Wikiversity. Wikiversity may also provide a useful forum for formal peer review.&quot;
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:26, 17 September 2020 (UTC)

Also at:
* [[Wikiversity:What is Wikiversity?#Wikiversity for researching]]

However:
* [[Wikiversity:Scope]]
*: &quot;The other kind of research is wiki-based original research. It is not yet clear that this will be part of the Wikiversity. If the Wikiversity community decides to support original research, it will have to develop a specific set of policies to support such research activities.&quot;

It seems Wikiversity:Scope needs an update to match the other pages. Alternatively, the page could be marked as archived and of historical interest only to ease maintenance burden of pages with overlapping scopes. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:38, 23 August 2023 (UTC)

== Your wikidebates on the Wikidebate homepage ==

Hi Dan! First, I'm amazed by the amount and quality of the debates, arguments and objections you produced last year. To be honest I didn't notice until recently, because I monitor activity from the [[Wikidebate#Recent changes|recent changes in the Wikidebate homepage]], but only changes to pages listed in the homepage are shown, so changes to your debates didn't show. Until now! I just added all of your debates to the homepage, so that should increase their visibility as well as the changes and additions done to them. Anyway, just thought you'd like to know. Again, amazing work, in the name of everyone who will be inspired, educated or interested by them, thanks!  [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 00:24, 11 January 2023 (UTC)
: It is very kind of you to say so and to increase the visibility of my work. Thank you very much. Last year, I was extremely enthusiastic about the debate format, as if possessed and driven by the ultimate spirit (&quot;enthusiasm&quot;). The debate format makes me a more honest thinker, being more ready to deal with the opposing arguments seriously. As a result of that enthusiasm, I tried to use the format and push it as far as I was able to, and I am planning to do more this year. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:15, 11 January 2023 (UTC)

== Would you like me to delete &quot;Is sharing personal images of oneself on social media a right that must be protected?&quot; ==

Would you like me to delete &quot;[[Is sharing personal images of oneself on social media a right that must be protected?]]&quot; &lt;big&gt;?&lt;/big&gt; -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:36, 12 January 2023 (UTC)
: You may delete the page if you wish, if you ask me. Nonetheless, for me, a redirect is as good as a deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:59, 12 January 2023 (UTC)

==A little praise to mathematics==
(A blog post.)

I heard the following two-line conversation between a German mathematician F. and a Chinese colleague Y.:
* Y: F., where have you learned to speak German so well?
* F: In mathematics.
What the above means is that in mathematics one learns to think in a certain way that leads to increased care about accuracy (true or false) and precision (broad or narrow concept) of one's formulation, of one's choice of words and concepts, etc. One rejects the so-called ''interpretation'' by which the interpreter is allowed to add words and modify words in a sentence and thereby as if interpret it. A genuine interpretation is the assignment of plain-meaning semantics to words, phrases, clauses and sentences; or there is also a genuine metaphorical interpretation; but adding words that the formulator forgot to state is no interpretation proper. The notion seems plausible enough.

I further heard F. say:
* A freshly graduated mathematician is someone who knows nothing and can learn anything.
That, clearly, is a hyperbole; anything refers to intellectual and cognitive endeavors, not, say, dancing or ice-skating. And not all cognitive enterprises can probably be penetrated or conquered (two different metaphors, but both metaphors) with the use of the tools of mathematics by any mathematician given his or her talent. But the general tenor stands: a freshly graduated mathematician has enough talent (innate gift) to practice mathematics and other rigorous thought, and enough tools of mathematics (cultural gifts or artifacts) to practice rigorous thought with. What a beautifully simple answer to point to a powerful idea.

As an aside, Y is a female and nominally did not contribute anything to the statements revealed. And yet, without her, F. would have never said the statement to the effect of, I learned to speak German well in mathematics. What Y did is what could possibly be a contribution more typical of females than males, namely asking questions and eliciting answers from males. One only has to think of the popular Slovak TV presenter Adela Banášová/Vinczeová and notice the remarkable talent for asking questions, quite possibly a typically feminine tool or weapon (both metaphors; what is the non-metaphor? Anyone?) However, this stereotyping of what is feminine requires a proper formal verification, and remains on the level of unproven hypothesis.

On a vaguely related note, I heard an Equador-American colleague J. say something like the following:
* J: When one properly masters the literal tools of language, one can better appreciate the metaphorical tools.
I did not fully appreciate the value of this back then, but I am starting to see ever better what he had in mind. The relation of mathematics to literalism is that, in a sense, tools of mathematics as tools of description are even more ''literalist'' or ''explicitist'' than the tools of non-mathematical formal language.

That was today's little Chautauqua, &quot;to edify and entertain, improve the mind and bring culture and enlightenment to the ears and thoughts of the hearer.&quot; (Greetings to Pirsig, a master of the phrase, formulation and metaphor.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:06, 7 February 2023 (UTC)

== [[Talk:Transgenderism - Polansky]] ==

Hello, I just want you to know that there is a discussion about a page that you made. Please join the discussion if possible. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:14, 27 July 2023 (UTC)

== Ethics of infanticide ==
In a preparation for a debate elsewhere, trying to avoid overburdening that debate, I will collect some reasoning concerning the topic of &quot;ethics of infanticide&quot;. It relates to the question whether a debate like [[Should infanticide be legal?]] should be allowed.

My contentions are the following:
* 1) &quot;Ethics of infanticide&quot; is a recognized academic subject, per https://philpapers.org/browse/infanticide.
* 2) The topic is of academic interest only in so far as some hold that infanticide is sometimes legitimate, or that arguments for that position should be explored even if one disagrees with the arguments.
* 3) One should not fool oneself into thinking that ethics is a nice and palatable subject. From my experience, a serious examination of the field of ethics leads to examining highly unpalatable questions, propositions and arguments.
* 4) Example academic article: [https://philpapers.org/rec/TEDDPF Dutch Protocols for Deliberately Ending the Life of Newborns: A Defence], philpapers.org
* 5) It is not clear why the debate format with arguments for and arguments against, with objections raised against arguments and objections raised against objections, is necessarily a worse or morally more objectionable format than a philosophical monologue in an article like [[Ethics of infanticide]].
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:31, 1 August 2023 (UTC)

==Ability of editors to reach decisions via votes==
The English Wikiversity does seem to have enough editors able to make decisions via votes:
* [[Wikiversity:Candidates for Custodianship/Koavf 2]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:46, 7 September 2023 (UTC)

== Deletion of [[Lexical unit]] ==

[[Lexical unit]] has been nominated for deletion.  Are you OK with that? [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:55, 15 October 2023 (UTC)
: @[[User:Guy vandegrift]]: Thank you for notifying me so that I can respond. In this case, I defer to other editors since: on the one hand, the page contains a minimum usable content: a definition and a good further reading, which is more than may non-deleted pages can say; on the other hand, the content is so small, is not an article but something very stubby, that it seems rational enough for the project to want to delete the page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:22, 16 October 2023 (UTC)

== Debates on policies of other Wikimedia projects ==

You've recently created the following debates regarding policy decisions on other Wikimedia projects:

* [[Should Wiktionary avoid use of straw polls?]]
* [[Should Wikipedia essays be moved out of Wikipedia namespace?]]
* [[Should Wiktionary user signatures be required to be unadorned default?]]
* [[Should Wiktionary votes cast be required to have a rationale?]]
* [[Should Wiktionary avoid indefinite blocks of productive users?]]
* [[Should Wiktionary have entries for inflected forms?]]
* [[Should Wiktionary require that all its information artifacts are sourced from reliable sources?]]
* [[Should Wiktionary use images?]]
* [[Is 60 percent a good threshold for Wikipedia consensus?]]
* [[Is Wikipedia consensus process good?]]

I'm concerned about these debates for two primary reasons:

# It's not clear that these policies involve topics which are within the educational scope of Wikiversity.
# Posting these debates here, rather than on the relevant project wikis, could be seen as an attempt by Wikiversity to interfere with policy discussions on those projects. For [[Wikiversity:Community Review/Wikimedia Ethics:Ethical Breaching Experiments|historical reasons]], this is a sensitive issue.

The latter is particularly troubling given your current block on the English Wiktionary.

[[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 22:16, 20 November 2023 (UTC)
: I read the above message and gave it some thought. I fail to see a serious problem, although some doubt is perhaps in order.
: As for the linked [[Wikiversity:Community Review/Wikimedia Ethics:Ethical Breaching Experiments]], the page documents some 2010 affair. I find the page and its subpages confusing; in any case, I am left confused. It must have been some very serious matter since there, Jimbo Wales threatened to close Wikiversity. Here is an [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Wikimedia_Ethics:Ethical_Breaching_Experiments&amp;oldid=548629 old revision], as a single page.
: If administrators see a problem, let me know, and let us determine whether some of the debates need to be deleted and why. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:51, 9 December 2023 (UTC)

== Re. [[Game Duenix for Amiga computers]] ==

Are you the author of this game, or do you have any evidence that this game was released by its author under a free license? While it may have been freely distributed during its lifetime in the 1990s, we would need an explicit release by the author under CC-BY-SA or a compatible license to host it as a learning resource on Wikiversity. [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 18:51, 20 December 2023 (UTC)
: I am the author. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:03, 20 December 2023 (UTC)
::Oh, neat! Never mind then. :) [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 19:12, 20 December 2023 (UTC)

== Wikidebate stuff ==

You've undone three of my contributions now. Why? You seem sore about something. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 14:34, 29 December 2023 (UTC)
: I hate to do it since it feels like censorship and I hate censorship, but it seems appropriate. I always try to explain in the edit summary what I am doing and why. I propose we discuss individual cases on the talk page of the respective debates, which seems to be the proper venue. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:37, 29 December 2023 (UTC)

== I read with great speed but my accuracy is sometimes off ==

My obsession with getting students to write essays on caused me to think &quot;Wikiversity&quot; when I looked at &quot;Wikipedia&quot; in your title: ''When I skimmed &quot;[[Should Wikipedia essays be moved out of Wikipedia namespace?]]''.  Nothing I said is relevant to your actual question. OOPS! [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:57, 12 February 2024 (UTC)

== Check redirect ==

Dan, when I processed [[Creating Examples of possible additional questions to ask the citrus grower]], I almost made the same mistake I often made: This page had no meaning until on realizes that it was a question asked in a (medium to low quality) page on Graphic Design.  In this hypothetical case, a graphic designer is working for a Citrus Growers Organization. At some point in our joint procedure, we need to check what links here.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:19, 23 February 2024 (UTC)
: Apologies for any confusion. I now checked [[Special:WhatLinksHere/Examples_of_possible_additional_questions_to_ask_the_citrus_grower]] and it is only linked from [[Wikiversity talk:Stubs]]; was it previously linked from somewhere else? I see the page is now at [[Graphic Design/Design Process/Problem Definition/Examples of possible additional questions to ask the citrus grower]], but [[Special:WhatLinksHere/Graphic_Design/Design_Process/Problem_Definition/Examples_of_possible_additional_questions_to_ask_the_citrus_grower]] finds nothing, so how is the page used, given nothing seems to refer to it? Thanks. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:23, 23 February 2024 (UTC)
: I need to pay attention better: we are talking ''redirect''. To avoid further confusion, I now substed the page at [[Graphic Design/Design Process/Problem Definition]] and renominated for deletion: I see no reason for a separate, deeply nested page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:30, 23 February 2024 (UTC)
::Copy-pasting was a good move.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:34, 23 February 2024 (UTC)

On a slightly different topic: On the Colloquium you proposed changes to [[Wikiversity:Deletions]].  I tried and failed to launch a focused discussion among a critical mass of active editors with [[Wikiversity:Deletion Convention 2024]].  It's not that we have no active and competent editors, but that they are all busy with other projects and not very interested.  You (with my help) are making radical changes in our policy.  I know that because we are deleting/moving custodians, curators, and bureaucrats have edited in the past.  For that reason, my guiding principle is that everything we do must be easily reversed.  Moving pages to userspace and [[Draft:Archive]] accomplishes that to my satisfaction.  As I go through the actual page-moves, I seem to prefer moving to userspace when there is an unambiguous single author.  I like to put a &lt;nowiki&gt;{{subst:prod}}&lt;/nowiki&gt; on pages I (we) need to think about. Is that OK with you?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:47, 23 February 2024 (UTC)
: I am generally happy with your manner of procedure. Where there is disagreement, we have a debate, and that's fine.
: About {{tlx|prod}}: I find {{tlx|rfd}} preferable over {{tlx|prod}}, but the latter is okay: it defers the deletion by multiple months, but that is tolerable. I think that deferral is usually not ideal, especially for pages that have ''not'' been created recently, but it does the work eventually. For freshly created pages, {{tlx|prod} pages sense, as in, give authors chance to expand their material; but then, moving to Draft: would also work, since the author could expand the material there and when it is more than sub-minimal, move it back to mainspace.
: About me &quot;making radical changes in our policy&quot;, I follow [[WV:Deletions]] and its phrase &quot;learning outcomes are scarce&quot;. So I do not see changes to policy. My nominations may deviate from recent practice in that I am applying [[WV:Deletions]] more rigorously than has before been the case. But then, a page with almost no statements and almost no further reading meets &quot;learning outcomes are scarce&quot;, has no saving graces, and should IMHO be moved out of mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:19, 23 February 2024 (UTC)

== Striking your words and creating space for voting ==
It was late when I struck your words regarding my motive for creating Draft:Archive.  I struck it because I '''thought''' it misrepresented something only I can know, which is my motive for doing something.  But it turns out that your words were correct (I often misread statements to be the opposite of what they actually say.)...Now about voting space:  I delete/move lots of pages, and I need to see a summary of where the community ''currently'' stands on each page.  For me, it is better if people deleted old votes and wrote in new votes (that magic word &quot;consensus&quot; can only become reality if people change their &quot;votes&quot;.)  On the other hand, I think we agree that &quot;discussion space&quot; is a place where the record needs to be kept (without modifications.)  To me, &quot;discussion space&quot; is like a loud bar or restaurant where people are talking simultaneously. I need quick summaries of where people currently stand so I can decide when to delete/move and when to close the discussion.  Feel free to express your discontent, because it is essential that all those who wish to change these rules are free to express their discontent.  If another person (or persons) shares your dissatisfaction with the &quot;voting space&quot; rules, we can and should hold a discussion on the topic....One more thing: I recently extended to requested maximum length for the &quot;voting&quot; section two allow for more nuanced positions.  Maybe that will help.  If you want, I can create a sub-sub section (with an extra = sign) with each person's name, so you can have more space to express your &quot;final&quot; position.  Would that help? --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandeg rift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:03, 10 March 2024 (UTC)
: The English Wikipedia and the English Wiktionary succeed in administering WP:AFD/WT:RFD without dedicated vote sections in their processes. I don't see why the English Wikiversity should not succeed in doing the same, provided editors learn to state e.g. '''keep''', '''delete''', '''leaning to keep''', '''leaning to delete''', etc. in boldface as part of their discussion contributions to make consensus determination easier. I think it key to emphasize the role of the strength of the argument standing in contrast to purely numerical consensus, and to allow something like the conjectures and refutations process, which can only work if refutations are allowed. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:51, 11 March 2024 (UTC)
::You ask why Wikiversity &quot;''should not'' (avoid a voting section)&quot;, the question is whether we &quot;''want to'' avoid a voting section.  That is how we did things in the past (see [[special:permalink/2612760]].) Why would I want to change?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:30, 15 March 2024 (UTC)
::: I am saying that specifically RFD should not have a &quot;Voting&quot; section (whereas your link is to [[Wikiversity talk:Drafts]]); it should be more like WP:AFD and WT:RFD. And if it has a voting section, responses to those &quot;votes&quot; should be allowed. That is not to say that there should be no element of voting in RFD, but rather that it should be a discussion in which vote positions are indicated via boldface. These &quot;shoulds&quot; represent my views and I recognize others do not need to agree, and my views may not necessarily prevail.  --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:25, 16 March 2024 (UTC)
::: To add clarity to my position: I have no qualms at all about how [[Wikiversity:Requests_for_Deletion#Facilitation]] is proceeding: people are posting their rationales and boldface stances/positions, but ''there is no separate heading &quot;Voting&quot;''. I do like that it is easy to determine numerical consensus from the boldface stances/positions. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:00, 16 March 2024 (UTC)
::::I like that idea. It makes it easier to change your &quot;vote&quot;: Simply unbold the old and boldface the new.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:18, 31 March 2024 (UTC)

==Google abandoned simple HTML version of Gmail==
(A blog post.) The simple HTML version of Gmail loaded instantly, was a pleasure to use and the visual design including colors looked great. The &quot;new and improved&quot; (not!) version takes several seconds to load and is rather displeasing. Oh well. One can read more in various online magazines. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:11, 18 March 2024 (UTC)

== Alternative to dewikifying categories ==

I noticed that you are removing &quot;what links here&quot; by removing category statements and such.  That is a good thing to do, but I was wondering if there is a better way to do it.  What if we put a backup copy into the history by copy/deleting the page, hitting &quot;save&quot; and then pasting the wikitext on the blank page. &lt;nowiki&gt;Then we get a bot to replace all instances of ]] and }} by ]*] and }*}.&lt;/nowiki&gt;  Some of the text will be corrupted, but the reader can just open the history and see the uncorrupted form.  I don't know much about bots, but perhaps the bot could also do the copy paste to create the uncorrupted version at the top of the history.

I added a search feature on [[Draft:Archive]] so that people could search the top (corrupted page) and would see enough to know whether they want to go into the history to read an uncorrupted version.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:10, 22 March 2024 (UTC)
: I only commented out category markup, thereby removing the page from the categories ([[Special:Diff/2614826|diff]]). I am not clear about why removing links and templates is in general necessary. Links to Wikipedia do not even appear in &quot;what links here&quot; of anything, AFAIK, so deactivating them seems even less useful than deactivating links within Wikiversity. Looking e.g. at [[Draft:Archive/2024/Openness]], the page now appears sort of broken; is this worth the objective of deactivating the links and template usage? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:51, 23 March 2024 (UTC)
::{{ping|Dan Polansky}} Upon reflection, I agree that there is no immediate need to dewikify {&quot;purge&quot;) anything.  In the long run, the page count in mainspace will likely grow at a slower rate than the pagecount in draft-archive space (since it will be rare to take something out of the latter.)  There is a simple remedy: Each year, we purge one years worth of draft-archive, with that year being the current year minus X. In other words, if X=4, we purge 2024 pages in 2028. That should be more than enough to ensure that most of the pages you see on any given category are associated with mainspace (instead of draft-archive.) If and when draft-archive starts to clutter things up, people of the future can create bots to purge.  Organizing the pages in draft-archive space by year will facilitate these purges. Also, routine maintenance of categories will lead to purges of draft-archived pages as pages that don't belong in the category (I believe you mentioned something like that in the wikidebate.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:42, 31 March 2024 (UTC)
:::Oh, and one more thing: The degradation of purged pages in draft-archive space is not a great problem because the reader can go into the history and find the page immediately after it was moved to draft-archive.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:49, 31 March 2024 (UTC)

== Speedy delete ==

Dan, please, don't delete my slowly moving stub. [[User:Janosabel|Janosabel]] ([[User talk:Janosabel|discuss]] • [[Special:Contributions/Janosabel|contribs]]) 16:36, 16 April 2024 (UTC)
: The above is very likely in reference to page [[Decentralized education]], which I originally marked for speedy deletion, but which is now in the ''3-month-deferred'' deletion process. According to that process, the page will be deleted only after 3 months, and even then, it may be moved to user space rather than outright deleted. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:43, 16 April 2024 (UTC)
::Many thanks for the respite. Yes, my request did relate to that stub. [[User:Janosabel|Janosabel]] ([[User talk:Janosabel|discuss]] • [[Special:Contributions/Janosabel|contribs]]) 18:24, 16 April 2024 (UTC)

== History of programming languages ==

Could you write an essay about the evolution from machine code to high-level programming languages? A history with some philosophical considerations? [[Special:Contributions/62.235.226.186|62.235.226.186]] ([[User talk:62.235.226.186|discuss]]) 20:27, 29 April 2024 (UTC)
: It's kind of you to ask. I am afraid I have a dearth of interesting ideas on the subject. But let me try: they say in jest that C is a portable assembly. If one learns assembly, one will learn how far that statement is from truth, even metaphorically. The jocular statement can at best be accepted as a mnemonic pointing to the fact that a reader of C code has it easy to imagine how blocks of bytes are being manipulated and what kind of assembly instructions are being generated by the compiler. Thus, compared to assembly, even the relatively low-level C--created in the 1970s and rather old now--is a very high level language that gives the power to control the behavior of the universal computing/typographic-manipulation machine into the hands of the masses, who are not ready or willing to learn assembly. Python goes farther in that direction (power to the masses) by being very legible, coming with batteries included (great standard library), and making it very easy to install permissively licensed 3rd party libraries (e.g. &quot;pip install numpy&quot;), of which there are many.
: The case of C makes it clear that the jump from assembly to a C-like language in its expressive power (C, procedural Pascal, procedural Basic, etc.) is a huge one (in bridging the gap between man and the machine), whereas the further jump to object-oriented languages is a much smaller one. To this day, C is one of the most important and widely used languages on the planet, used by operating system kernels, CPython, Git, Gtk and GIMP, etc. One would thus think that by 1970s, the most important programming language inventions were already made. However, I am only a single person with a limited experience and other people could persuasively argue that object-oriented programming is in fact a big deal, found in C++, Python and Java, which together with C dominate the Tiobe index. I do not deny the value of OOP in the domains where it is most fit for use, but it seems clear that the economic law of diminishing returns is at play and that the added value of procedural high-level programming over assembly is much larger than the added value of OOP. One could add managed runtimes (JVM, .NET) as a separate invention with a huge impact on reliability and cost reduction.
: I am no expert on programming language history. I unfairly omitted FORTRAN, with which I have no experience. I do not know to what extent the early FORTRAN was procedural and structured or whether it resembled the 8-bit Basics with plentiful use of goto; I would have to check. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:42, 3 May 2024 (UTC)

== Edifying fields of study ==

In your view, which disciplines / intellectual activities are the most edifying / mind-sharpening? Computer programming, mathematics, writing, learning to play a musical instrument, something else? [[Special:Contributions/62.235.226.186|62.235.226.186]] ([[User talk:62.235.226.186|discuss]]) 21:02, 29 April 2024 (UTC)</text>
      <sha1>l3p48ihss4sn9sj9lz34yuum0gra81d</sha1>
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      <id>2624951</id>
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        <username>Dan Polansky</username>
        <id>33469</id>
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      <comment>/* Edifying fields of study */</comment>
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      <text bytes="47359" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
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&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:12, 17 March 2020 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}

== COVID-19 ==

Thank you Dan for the support and contribution to the [[COVID-19]] learning resources. I appreciate your contributions very much. Would like to coordinate collaborative effort a bit due to the dynamic change of COVID-19 situation globally. Is there a specific subpage (not user-page) that is content driven, that you would like to add you expertise e.g. Data Analysis?. Shall we revise the structure to be more user-friendly for finding specific learning resources? Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 13:10, 17 March 2020 (UTC)
: Thank you. As for Wikiversity, I really do not know how things are working here. I have encouraged another editor to create [[COVID-19/Julian Mendez]], which is original research and is super interesting. However, I have not reviewed the material much, just had a superficial glance. On the surface, the thinking is good: it emphasizes time lag of detection of a rapidly exponentially increasing phenomenon.
: Since Wikiversity allows original research, it presents a unique opportunity for material like  [[COVID-19/Julian Mendez]]. My experience from data analysis is somewhat limited; I have a pretty strong mathematical background, so I know that the derivative of ''a^x'' is ''ln a * a^x'', and that gives a super scary light on the covid thing, how both totals and daily increases have the same base of exponential growth until mitigated, whether cases or deaths.
: I probably do not have the energy to coordinate efforts and I have no improvement proposals on [[COVID-19]] structure and content. I am wondering when I am finally going to run out of gumption. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 17 March 2020 (UTC)

:: Thank you, no worries. Exponential growth might not be the appropriate mathematical model to describe the development of COVID-19 or an epidemiological outbreak in general, because developement of count has limits of growth e.g. the total population on earth. So logistical growth with a capacity is more likely to describe the development. Currently in the early phase the data shows an exponential pattern but closer to the capacity the derivation gets smaller and closer to zero. The question is, what is the capacity of the logistical growth. Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:24, 17 March 2020 (UTC)
::: The initial phase is exponential, and this is easily empirically verified by observing the straight lines in the graphs with logarithmic y-axis, but it is true that once factors limiting the growth set in, it ceases to be exponential. Without intervention/mitigation, starting to run out of people to infect is the main limiting factor of the exponential growth, from what I can see. However, this is where we do not want to get since that becomes numerically significant only after, say, 10% of the population gets infected, and luckily enough, hardly any country has come close to that degree. And this would be an interesting mathematical/epidemiological assignment for a classroom: determine at which degree of population penetration an unmitigated infection growth ceases significantly to be exponential, for some value of &quot;significantly&quot;. That would need to assume some model of spread; I was thinking of molecules in a gas hitting one another, but the social phenomenon may look much different because of non-Gaussian distribution of &quot;influencers&quot;, as it were; there would be some people who have hugely many contacts and targetting them specifically for isolation could make huge difference, and you do not get that in a gas, I suppose. But the gas model need not be so bad to get a very first idea, and isolating &quot;influencers&quot; would not need to suffice at all. I don't really know, but I do maintain that the virus growth is in an exponential phase and, unless mitigated, would stay in the exponential phase in the coming weeks in most countries that would not do mitigation. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:40, 17 March 2020 (UTC)
: For reference, I keep on expanding the following pages: [[COVID-19/Dan Polansky]] and [[Talk:COVID-19/Dan Polansky]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:39, 3 May 2020 (UTC)
: ''Late retraction'': As for &quot;The initial phase is exponential, and this is easily empirically verified by observing the straight lines in the graphs with logarithmic y-axis, but it is true that once factors limiting the growth set in, it ceases to be exponential&quot;: that is wrong or misleading; while the confirmed cases were indeed originally growing exponentially, the observed rate of exponential growth was due to exponential growth in number of tests, as is confirmed by observing test positivity rate and observing the rate of growth of tests. The true infection count could either be initially growing exponentially at hugely slower rate than the nominal confirmed cases, or they were growing according to Gompertz curve (see research by Michael Levitt) and therefore never growing exponentially. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:28, 15 August 2020 (UTC)

=== Effiency of Lock Down ===
thank you for adding that important topic, to COVID-19 learning resource, Best regards, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:13, 23 July 2020 (UTC)

== Transgenderism ==

I am experimenting with the following page: [[User:Dan Polansky/Transgenderism]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:48, 17 July 2020 (UTC)

== COVID-19 Data ==

Is the COVID-19 data you've added available in Wikidata? It would be better to have the data there and query it rather than saved as pages here. As Wikidata, anyone could query it in any language and for any Wikimedia project. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:33, 14 August 2020 (UTC)
: Not that I know of. I see your point with cross-wiki query and avoidance of duplication of storage. On the other hand, storing comma-separated lists of values directly in the wiki markup is very simple and convenient, and one can very easily take that and calculate e.g. moving averages from that. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:13, 15 August 2020 (UTC)

::This gets a little tricky, because [[Wikiversity:What Wikiversity is not|Wikiversity does not]] duplicate other Wikimedia projects. I haven't done a large data import and query like this, but I'm willing to try one and see if I can produce the same results using the preferred structure. Is there a particular page and data set you would recommend as a starting point? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:47, 15 August 2020 (UTC)
::: You may try [[COVID-19/All-cause deaths/London]] if you wish; it uses sources different from the other pages. Let me note that I am not very enthusiastic; I am afraid of making simple things more complex at just a little benefit. If, say, I will want to update the London data as I did today, instead of doing something utterly simple and straightforward I will either need to ask you for help or learn about Wikidata imports myself. The loss of personal productivity is likely to be non-trivial and may be a showstoper for me; instead of updating London, I would then do something more straightfoward. And when I learn how to do Wikidata and someone else wants to update London, it is now them who has to learn Wikidata.
::: What would be really useful would be to update the charting add-in in Wikiversity to be on par with Wikipedia: in Wikipedia, it produces raster images whereas in Wikiversity, it produces some kind of semi-live object that seems to take longer to load. (I misspoke; the thing produced in Wikiversity is a PNG image as well; it does not have antialised fonts in the x-axis when the labels are at angle. Wikipedia seems more up to date with the add-in.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:03, 15 August 2020 (UTC)

::::I've updated [[Module:Graph]]. If that's not it, you'll need to be more specific as to what is more current at Wikipedia so it can be imported. You can also request imports yourself at [[Wikiversity:Import]]. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:17, 15 August 2020 (UTC)
::::: Thank you! It did not help: the x-axis fonts are still not antialiased and there is still a change to red color on mouseover over the blue line. [[Module:Graph]] is only a layer over the extension itself ([https://www.mediawiki.org/wiki/Extension:Graph Extension:Graph]), and maybe the extension needs an update. Theoretically, [[Template:Graph:Chart]] might need an update as well. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:28, 16 August 2020 (UTC)

::::::See [[Special:Version]]. Extension:Graph appears to be current. I checked the dates on Template:Graph:Chart initially and it was also current. I'm happy to import whatever we need to update, but I'll need your help to find whatever that might be. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:43, 16 August 2020 (UTC)
: (Outdent) Let me double check:
:* Per [[Special:Version]], Extension:Graph: Wikiversity: (9e762ac) 06:27, 6 August 2020; Wikipedia: (9e762ac) 06:27, 6 August 2020
:* [[Module:Graph]], textual comparison between WV and WP: same
:* [[Template:Graph:Chart]], textual comparison between WV and WP: same
:* [[Template:Graph:Chart/styles.css]], textual comparison between WV and WP: same except for a comment line, immaterial
:* Raw test of the JSON markup in [[User:Dan Polansky/sandbox]]: no x-axis antialiasing
: Could there be a relevant setting at LocalSettings.php? [https://www.mediawiki.org/wiki/Extension_talk:Graph Mediawiki's Extension talk:Graph] mentions $wgGraphImgServiceUrl, so something like that. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:34, 17 August 2020 (UTC)

::At this point, the best option would be to file a [[phabricator:]] ticket and see if one of the developers can identify the problem. We can't make php setting changes, so the ticket will be necessary anyway. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:26, 17 August 2020 (UTC)
::: It seems that on Wikiversity, the charts are made using ''canvas'' element, and they are plotted by the client browser via Javascript. By contrast, the English Wikipedia seems to be customized to have a server backend generate the PNG images for the charts so the client browser does not have to do any plotting, showin an ''img'' element instead. One consequence is that Wikipedia charts show fine on older devices whose browsers do not support canvas element. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:14, 8 September 2020 (UTC)

== Original research ==

For my reference:
* [[Wikiversity:Original research]]
*: &quot;Original research which meets the guidelines of this policy is permitted on Wikiversity. Researchers devoted to scholarly investigation using sound, ethical methods are encouraged to develop and disseminate their work via Wikiversity. Wikiversity may also provide a useful forum for formal peer review.&quot;
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:26, 17 September 2020 (UTC)

Also at:
* [[Wikiversity:What is Wikiversity?#Wikiversity for researching]]

However:
* [[Wikiversity:Scope]]
*: &quot;The other kind of research is wiki-based original research. It is not yet clear that this will be part of the Wikiversity. If the Wikiversity community decides to support original research, it will have to develop a specific set of policies to support such research activities.&quot;

It seems Wikiversity:Scope needs an update to match the other pages. Alternatively, the page could be marked as archived and of historical interest only to ease maintenance burden of pages with overlapping scopes. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:38, 23 August 2023 (UTC)

== Your wikidebates on the Wikidebate homepage ==

Hi Dan! First, I'm amazed by the amount and quality of the debates, arguments and objections you produced last year. To be honest I didn't notice until recently, because I monitor activity from the [[Wikidebate#Recent changes|recent changes in the Wikidebate homepage]], but only changes to pages listed in the homepage are shown, so changes to your debates didn't show. Until now! I just added all of your debates to the homepage, so that should increase their visibility as well as the changes and additions done to them. Anyway, just thought you'd like to know. Again, amazing work, in the name of everyone who will be inspired, educated or interested by them, thanks!  [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 00:24, 11 January 2023 (UTC)
: It is very kind of you to say so and to increase the visibility of my work. Thank you very much. Last year, I was extremely enthusiastic about the debate format, as if possessed and driven by the ultimate spirit (&quot;enthusiasm&quot;). The debate format makes me a more honest thinker, being more ready to deal with the opposing arguments seriously. As a result of that enthusiasm, I tried to use the format and push it as far as I was able to, and I am planning to do more this year. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:15, 11 January 2023 (UTC)

== Would you like me to delete &quot;Is sharing personal images of oneself on social media a right that must be protected?&quot; ==

Would you like me to delete &quot;[[Is sharing personal images of oneself on social media a right that must be protected?]]&quot; &lt;big&gt;?&lt;/big&gt; -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:36, 12 January 2023 (UTC)
: You may delete the page if you wish, if you ask me. Nonetheless, for me, a redirect is as good as a deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:59, 12 January 2023 (UTC)

==A little praise to mathematics==
(A blog post.)

I heard the following two-line conversation between a German mathematician F. and a Chinese colleague Y.:
* Y: F., where have you learned to speak German so well?
* F: In mathematics.
What the above means is that in mathematics one learns to think in a certain way that leads to increased care about accuracy (true or false) and precision (broad or narrow concept) of one's formulation, of one's choice of words and concepts, etc. One rejects the so-called ''interpretation'' by which the interpreter is allowed to add words and modify words in a sentence and thereby as if interpret it. A genuine interpretation is the assignment of plain-meaning semantics to words, phrases, clauses and sentences; or there is also a genuine metaphorical interpretation; but adding words that the formulator forgot to state is no interpretation proper. The notion seems plausible enough.

I further heard F. say:
* A freshly graduated mathematician is someone who knows nothing and can learn anything.
That, clearly, is a hyperbole; anything refers to intellectual and cognitive endeavors, not, say, dancing or ice-skating. And not all cognitive enterprises can probably be penetrated or conquered (two different metaphors, but both metaphors) with the use of the tools of mathematics by any mathematician given his or her talent. But the general tenor stands: a freshly graduated mathematician has enough talent (innate gift) to practice mathematics and other rigorous thought, and enough tools of mathematics (cultural gifts or artifacts) to practice rigorous thought with. What a beautifully simple answer to point to a powerful idea.

As an aside, Y is a female and nominally did not contribute anything to the statements revealed. And yet, without her, F. would have never said the statement to the effect of, I learned to speak German well in mathematics. What Y did is what could possibly be a contribution more typical of females than males, namely asking questions and eliciting answers from males. One only has to think of the popular Slovak TV presenter Adela Banášová/Vinczeová and notice the remarkable talent for asking questions, quite possibly a typically feminine tool or weapon (both metaphors; what is the non-metaphor? Anyone?) However, this stereotyping of what is feminine requires a proper formal verification, and remains on the level of unproven hypothesis.

On a vaguely related note, I heard an Equador-American colleague J. say something like the following:
* J: When one properly masters the literal tools of language, one can better appreciate the metaphorical tools.
I did not fully appreciate the value of this back then, but I am starting to see ever better what he had in mind. The relation of mathematics to literalism is that, in a sense, tools of mathematics as tools of description are even more ''literalist'' or ''explicitist'' than the tools of non-mathematical formal language.

That was today's little Chautauqua, &quot;to edify and entertain, improve the mind and bring culture and enlightenment to the ears and thoughts of the hearer.&quot; (Greetings to Pirsig, a master of the phrase, formulation and metaphor.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:06, 7 February 2023 (UTC)

== [[Talk:Transgenderism - Polansky]] ==

Hello, I just want you to know that there is a discussion about a page that you made. Please join the discussion if possible. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:14, 27 July 2023 (UTC)

== Ethics of infanticide ==
In a preparation for a debate elsewhere, trying to avoid overburdening that debate, I will collect some reasoning concerning the topic of &quot;ethics of infanticide&quot;. It relates to the question whether a debate like [[Should infanticide be legal?]] should be allowed.

My contentions are the following:
* 1) &quot;Ethics of infanticide&quot; is a recognized academic subject, per https://philpapers.org/browse/infanticide.
* 2) The topic is of academic interest only in so far as some hold that infanticide is sometimes legitimate, or that arguments for that position should be explored even if one disagrees with the arguments.
* 3) One should not fool oneself into thinking that ethics is a nice and palatable subject. From my experience, a serious examination of the field of ethics leads to examining highly unpalatable questions, propositions and arguments.
* 4) Example academic article: [https://philpapers.org/rec/TEDDPF Dutch Protocols for Deliberately Ending the Life of Newborns: A Defence], philpapers.org
* 5) It is not clear why the debate format with arguments for and arguments against, with objections raised against arguments and objections raised against objections, is necessarily a worse or morally more objectionable format than a philosophical monologue in an article like [[Ethics of infanticide]].
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:31, 1 August 2023 (UTC)

==Ability of editors to reach decisions via votes==
The English Wikiversity does seem to have enough editors able to make decisions via votes:
* [[Wikiversity:Candidates for Custodianship/Koavf 2]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:46, 7 September 2023 (UTC)

== Deletion of [[Lexical unit]] ==

[[Lexical unit]] has been nominated for deletion.  Are you OK with that? [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:55, 15 October 2023 (UTC)
: @[[User:Guy vandegrift]]: Thank you for notifying me so that I can respond. In this case, I defer to other editors since: on the one hand, the page contains a minimum usable content: a definition and a good further reading, which is more than may non-deleted pages can say; on the other hand, the content is so small, is not an article but something very stubby, that it seems rational enough for the project to want to delete the page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:22, 16 October 2023 (UTC)

== Debates on policies of other Wikimedia projects ==

You've recently created the following debates regarding policy decisions on other Wikimedia projects:

* [[Should Wiktionary avoid use of straw polls?]]
* [[Should Wikipedia essays be moved out of Wikipedia namespace?]]
* [[Should Wiktionary user signatures be required to be unadorned default?]]
* [[Should Wiktionary votes cast be required to have a rationale?]]
* [[Should Wiktionary avoid indefinite blocks of productive users?]]
* [[Should Wiktionary have entries for inflected forms?]]
* [[Should Wiktionary require that all its information artifacts are sourced from reliable sources?]]
* [[Should Wiktionary use images?]]
* [[Is 60 percent a good threshold for Wikipedia consensus?]]
* [[Is Wikipedia consensus process good?]]

I'm concerned about these debates for two primary reasons:

# It's not clear that these policies involve topics which are within the educational scope of Wikiversity.
# Posting these debates here, rather than on the relevant project wikis, could be seen as an attempt by Wikiversity to interfere with policy discussions on those projects. For [[Wikiversity:Community Review/Wikimedia Ethics:Ethical Breaching Experiments|historical reasons]], this is a sensitive issue.

The latter is particularly troubling given your current block on the English Wiktionary.

[[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 22:16, 20 November 2023 (UTC)
: I read the above message and gave it some thought. I fail to see a serious problem, although some doubt is perhaps in order.
: As for the linked [[Wikiversity:Community Review/Wikimedia Ethics:Ethical Breaching Experiments]], the page documents some 2010 affair. I find the page and its subpages confusing; in any case, I am left confused. It must have been some very serious matter since there, Jimbo Wales threatened to close Wikiversity. Here is an [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Wikimedia_Ethics:Ethical_Breaching_Experiments&amp;oldid=548629 old revision], as a single page.
: If administrators see a problem, let me know, and let us determine whether some of the debates need to be deleted and why. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:51, 9 December 2023 (UTC)

== Re. [[Game Duenix for Amiga computers]] ==

Are you the author of this game, or do you have any evidence that this game was released by its author under a free license? While it may have been freely distributed during its lifetime in the 1990s, we would need an explicit release by the author under CC-BY-SA or a compatible license to host it as a learning resource on Wikiversity. [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 18:51, 20 December 2023 (UTC)
: I am the author. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:03, 20 December 2023 (UTC)
::Oh, neat! Never mind then. :) [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 19:12, 20 December 2023 (UTC)

== Wikidebate stuff ==

You've undone three of my contributions now. Why? You seem sore about something. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 14:34, 29 December 2023 (UTC)
: I hate to do it since it feels like censorship and I hate censorship, but it seems appropriate. I always try to explain in the edit summary what I am doing and why. I propose we discuss individual cases on the talk page of the respective debates, which seems to be the proper venue. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:37, 29 December 2023 (UTC)

== I read with great speed but my accuracy is sometimes off ==

My obsession with getting students to write essays on caused me to think &quot;Wikiversity&quot; when I looked at &quot;Wikipedia&quot; in your title: ''When I skimmed &quot;[[Should Wikipedia essays be moved out of Wikipedia namespace?]]''.  Nothing I said is relevant to your actual question. OOPS! [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:57, 12 February 2024 (UTC)

== Check redirect ==

Dan, when I processed [[Creating Examples of possible additional questions to ask the citrus grower]], I almost made the same mistake I often made: This page had no meaning until on realizes that it was a question asked in a (medium to low quality) page on Graphic Design.  In this hypothetical case, a graphic designer is working for a Citrus Growers Organization. At some point in our joint procedure, we need to check what links here.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:19, 23 February 2024 (UTC)
: Apologies for any confusion. I now checked [[Special:WhatLinksHere/Examples_of_possible_additional_questions_to_ask_the_citrus_grower]] and it is only linked from [[Wikiversity talk:Stubs]]; was it previously linked from somewhere else? I see the page is now at [[Graphic Design/Design Process/Problem Definition/Examples of possible additional questions to ask the citrus grower]], but [[Special:WhatLinksHere/Graphic_Design/Design_Process/Problem_Definition/Examples_of_possible_additional_questions_to_ask_the_citrus_grower]] finds nothing, so how is the page used, given nothing seems to refer to it? Thanks. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:23, 23 February 2024 (UTC)
: I need to pay attention better: we are talking ''redirect''. To avoid further confusion, I now substed the page at [[Graphic Design/Design Process/Problem Definition]] and renominated for deletion: I see no reason for a separate, deeply nested page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:30, 23 February 2024 (UTC)
::Copy-pasting was a good move.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:34, 23 February 2024 (UTC)

On a slightly different topic: On the Colloquium you proposed changes to [[Wikiversity:Deletions]].  I tried and failed to launch a focused discussion among a critical mass of active editors with [[Wikiversity:Deletion Convention 2024]].  It's not that we have no active and competent editors, but that they are all busy with other projects and not very interested.  You (with my help) are making radical changes in our policy.  I know that because we are deleting/moving custodians, curators, and bureaucrats have edited in the past.  For that reason, my guiding principle is that everything we do must be easily reversed.  Moving pages to userspace and [[Draft:Archive]] accomplishes that to my satisfaction.  As I go through the actual page-moves, I seem to prefer moving to userspace when there is an unambiguous single author.  I like to put a &lt;nowiki&gt;{{subst:prod}}&lt;/nowiki&gt; on pages I (we) need to think about. Is that OK with you?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:47, 23 February 2024 (UTC)
: I am generally happy with your manner of procedure. Where there is disagreement, we have a debate, and that's fine.
: About {{tlx|prod}}: I find {{tlx|rfd}} preferable over {{tlx|prod}}, but the latter is okay: it defers the deletion by multiple months, but that is tolerable. I think that deferral is usually not ideal, especially for pages that have ''not'' been created recently, but it does the work eventually. For freshly created pages, {{tlx|prod} pages sense, as in, give authors chance to expand their material; but then, moving to Draft: would also work, since the author could expand the material there and when it is more than sub-minimal, move it back to mainspace.
: About me &quot;making radical changes in our policy&quot;, I follow [[WV:Deletions]] and its phrase &quot;learning outcomes are scarce&quot;. So I do not see changes to policy. My nominations may deviate from recent practice in that I am applying [[WV:Deletions]] more rigorously than has before been the case. But then, a page with almost no statements and almost no further reading meets &quot;learning outcomes are scarce&quot;, has no saving graces, and should IMHO be moved out of mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:19, 23 February 2024 (UTC)

== Striking your words and creating space for voting ==
It was late when I struck your words regarding my motive for creating Draft:Archive.  I struck it because I '''thought''' it misrepresented something only I can know, which is my motive for doing something.  But it turns out that your words were correct (I often misread statements to be the opposite of what they actually say.)...Now about voting space:  I delete/move lots of pages, and I need to see a summary of where the community ''currently'' stands on each page.  For me, it is better if people deleted old votes and wrote in new votes (that magic word &quot;consensus&quot; can only become reality if people change their &quot;votes&quot;.)  On the other hand, I think we agree that &quot;discussion space&quot; is a place where the record needs to be kept (without modifications.)  To me, &quot;discussion space&quot; is like a loud bar or restaurant where people are talking simultaneously. I need quick summaries of where people currently stand so I can decide when to delete/move and when to close the discussion.  Feel free to express your discontent, because it is essential that all those who wish to change these rules are free to express their discontent.  If another person (or persons) shares your dissatisfaction with the &quot;voting space&quot; rules, we can and should hold a discussion on the topic....One more thing: I recently extended to requested maximum length for the &quot;voting&quot; section two allow for more nuanced positions.  Maybe that will help.  If you want, I can create a sub-sub section (with an extra = sign) with each person's name, so you can have more space to express your &quot;final&quot; position.  Would that help? --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandeg rift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:03, 10 March 2024 (UTC)
: The English Wikipedia and the English Wiktionary succeed in administering WP:AFD/WT:RFD without dedicated vote sections in their processes. I don't see why the English Wikiversity should not succeed in doing the same, provided editors learn to state e.g. '''keep''', '''delete''', '''leaning to keep''', '''leaning to delete''', etc. in boldface as part of their discussion contributions to make consensus determination easier. I think it key to emphasize the role of the strength of the argument standing in contrast to purely numerical consensus, and to allow something like the conjectures and refutations process, which can only work if refutations are allowed. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:51, 11 March 2024 (UTC)
::You ask why Wikiversity &quot;''should not'' (avoid a voting section)&quot;, the question is whether we &quot;''want to'' avoid a voting section.  That is how we did things in the past (see [[special:permalink/2612760]].) Why would I want to change?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:30, 15 March 2024 (UTC)
::: I am saying that specifically RFD should not have a &quot;Voting&quot; section (whereas your link is to [[Wikiversity talk:Drafts]]); it should be more like WP:AFD and WT:RFD. And if it has a voting section, responses to those &quot;votes&quot; should be allowed. That is not to say that there should be no element of voting in RFD, but rather that it should be a discussion in which vote positions are indicated via boldface. These &quot;shoulds&quot; represent my views and I recognize others do not need to agree, and my views may not necessarily prevail.  --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:25, 16 March 2024 (UTC)
::: To add clarity to my position: I have no qualms at all about how [[Wikiversity:Requests_for_Deletion#Facilitation]] is proceeding: people are posting their rationales and boldface stances/positions, but ''there is no separate heading &quot;Voting&quot;''. I do like that it is easy to determine numerical consensus from the boldface stances/positions. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:00, 16 March 2024 (UTC)
::::I like that idea. It makes it easier to change your &quot;vote&quot;: Simply unbold the old and boldface the new.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:18, 31 March 2024 (UTC)

==Google abandoned simple HTML version of Gmail==
(A blog post.) The simple HTML version of Gmail loaded instantly, was a pleasure to use and the visual design including colors looked great. The &quot;new and improved&quot; (not!) version takes several seconds to load and is rather displeasing. Oh well. One can read more in various online magazines. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:11, 18 March 2024 (UTC)

== Alternative to dewikifying categories ==

I noticed that you are removing &quot;what links here&quot; by removing category statements and such.  That is a good thing to do, but I was wondering if there is a better way to do it.  What if we put a backup copy into the history by copy/deleting the page, hitting &quot;save&quot; and then pasting the wikitext on the blank page. &lt;nowiki&gt;Then we get a bot to replace all instances of ]] and }} by ]*] and }*}.&lt;/nowiki&gt;  Some of the text will be corrupted, but the reader can just open the history and see the uncorrupted form.  I don't know much about bots, but perhaps the bot could also do the copy paste to create the uncorrupted version at the top of the history.

I added a search feature on [[Draft:Archive]] so that people could search the top (corrupted page) and would see enough to know whether they want to go into the history to read an uncorrupted version.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:10, 22 March 2024 (UTC)
: I only commented out category markup, thereby removing the page from the categories ([[Special:Diff/2614826|diff]]). I am not clear about why removing links and templates is in general necessary. Links to Wikipedia do not even appear in &quot;what links here&quot; of anything, AFAIK, so deactivating them seems even less useful than deactivating links within Wikiversity. Looking e.g. at [[Draft:Archive/2024/Openness]], the page now appears sort of broken; is this worth the objective of deactivating the links and template usage? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:51, 23 March 2024 (UTC)
::{{ping|Dan Polansky}} Upon reflection, I agree that there is no immediate need to dewikify {&quot;purge&quot;) anything.  In the long run, the page count in mainspace will likely grow at a slower rate than the pagecount in draft-archive space (since it will be rare to take something out of the latter.)  There is a simple remedy: Each year, we purge one years worth of draft-archive, with that year being the current year minus X. In other words, if X=4, we purge 2024 pages in 2028. That should be more than enough to ensure that most of the pages you see on any given category are associated with mainspace (instead of draft-archive.) If and when draft-archive starts to clutter things up, people of the future can create bots to purge.  Organizing the pages in draft-archive space by year will facilitate these purges. Also, routine maintenance of categories will lead to purges of draft-archived pages as pages that don't belong in the category (I believe you mentioned something like that in the wikidebate.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:42, 31 March 2024 (UTC)
:::Oh, and one more thing: The degradation of purged pages in draft-archive space is not a great problem because the reader can go into the history and find the page immediately after it was moved to draft-archive.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:49, 31 March 2024 (UTC)

== Speedy delete ==

Dan, please, don't delete my slowly moving stub. [[User:Janosabel|Janosabel]] ([[User talk:Janosabel|discuss]] • [[Special:Contributions/Janosabel|contribs]]) 16:36, 16 April 2024 (UTC)
: The above is very likely in reference to page [[Decentralized education]], which I originally marked for speedy deletion, but which is now in the ''3-month-deferred'' deletion process. According to that process, the page will be deleted only after 3 months, and even then, it may be moved to user space rather than outright deleted. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:43, 16 April 2024 (UTC)
::Many thanks for the respite. Yes, my request did relate to that stub. [[User:Janosabel|Janosabel]] ([[User talk:Janosabel|discuss]] • [[Special:Contributions/Janosabel|contribs]]) 18:24, 16 April 2024 (UTC)

== History of programming languages ==

Could you write an essay about the evolution from machine code to high-level programming languages? A history with some philosophical considerations? [[Special:Contributions/62.235.226.186|62.235.226.186]] ([[User talk:62.235.226.186|discuss]]) 20:27, 29 April 2024 (UTC)
: It's kind of you to ask. I am afraid I have a dearth of interesting ideas on the subject. But let me try: they say in jest that C is a portable assembly. If one learns assembly, one will learn how far that statement is from truth, even metaphorically. The jocular statement can at best be accepted as a mnemonic pointing to the fact that a reader of C code has it easy to imagine how blocks of bytes are being manipulated and what kind of assembly instructions are being generated by the compiler. Thus, compared to assembly, even the relatively low-level C--created in the 1970s and rather old now--is a very high level language that gives the power to control the behavior of the universal computing/typographic-manipulation machine into the hands of the masses, who are not ready or willing to learn assembly. Python goes farther in that direction (power to the masses) by being very legible, coming with batteries included (great standard library), and making it very easy to install permissively licensed 3rd party libraries (e.g. &quot;pip install numpy&quot;), of which there are many.
: The case of C makes it clear that the jump from assembly to a C-like language in its expressive power (C, procedural Pascal, procedural Basic, etc.) is a huge one (in bridging the gap between man and the machine), whereas the further jump to object-oriented languages is a much smaller one. To this day, C is one of the most important and widely used languages on the planet, used by operating system kernels, CPython, Git, Gtk and GIMP, etc. One would thus think that by 1970s, the most important programming language inventions were already made. However, I am only a single person with a limited experience and other people could persuasively argue that object-oriented programming is in fact a big deal, found in C++, Python and Java, which together with C dominate the Tiobe index. I do not deny the value of OOP in the domains where it is most fit for use, but it seems clear that the economic law of diminishing returns is at play and that the added value of procedural high-level programming over assembly is much larger than the added value of OOP. One could add managed runtimes (JVM, .NET) as a separate invention with a huge impact on reliability and cost reduction.
: I am no expert on programming language history. I unfairly omitted FORTRAN, with which I have no experience. I do not know to what extent the early FORTRAN was procedural and structured or whether it resembled the 8-bit Basics with plentiful use of goto; I would have to check. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:42, 3 May 2024 (UTC)

== Edifying fields of study ==

In your view, which disciplines / intellectual activities are the most edifying / mind-sharpening? Computer programming, mathematics, writing, learning to play a musical instrument, something else? [[Special:Contributions/62.235.226.186|62.235.226.186]] ([[User talk:62.235.226.186|discuss]]) 21:02, 29 April 2024 (UTC)
: I play guitar, but I do not find it mind-sharpening. Other than that, I find all of computer programming, mathematics and writing mind-sharpening. Among them, computer programming is the only clearly Popperian/empirical-refutation teacher: one is being refuted again and again, one makes mistakes without being able to argue one's way out of them, one forms hypotheses and finds them corrected/refined by testing. By contrast, paper is very patient (as for math and writing) and mistakes in one's math or writing seem much easier to go undetected. On the other hand, computer programming in no way replaces doing math and writing; these activities require and develop skills that seem to be to a large extent complementary. Moreover, one can approach writing in a Popperian spirit: write hypotheses down and try to refute them or find arguments against them. And even if what one is writing down seems to be clearly true rather than conjectural, one can try to play the devil's advocate against one's position anyway. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:45, 3 May 2024 (UTC)</text>
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      <text bytes="14838" xml:space="preserve">== Also Known As ==
*Family name
**Blackwood (2 September 1862)&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|date=2021-05-20|title=Frederick Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava&amp;oldid=1024107132|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava.&lt;/ref&gt;
**Hamilton-Blackwood (2 September 1862)&lt;ref name=&quot;:0&quot;&gt;&quot;Frederick Temple Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava.&quot; {{Cite web|url=https://www.thepeerage.com/p4999.htm#i49984|title=Person Page|website=www.thepeerage.com|access-date=2021-05-31}} https://www.thepeerage.com/p4999.htm#i49984.&lt;/ref&gt;
**Hamilton-Temple-Blackwood (13 November 1872)
*Earl of Dufferin, co. Down (U.K. peerage, created 1871)&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2021-10-19|title=Baron Dufferin and Claneboye|url=https://en.wikipedia.org/w/index.php?title=Baron_Dufferin_and_Claneboye&amp;oldid=1050709226|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Baron_Dufferin_and_Claneboye.&lt;/ref&gt;
**Frederick Temple Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava (13 November 1871 – 12 February 1902)
*Marquess of Dufferin and Ava (U.K. peerage, created 1888)&lt;ref name=&quot;:5&quot; /&gt;
**Frederick Temple Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava (17 November 1888 – 12 February 1902)
*Earl of Ava, co. Down and Burma (U.K. peerage, created 1888)&lt;ref name=&quot;:5&quot; /&gt;
**Frederick Temple Hamilton-Temple-Blackwood, 1st? Earl of Ava (17 November 1888 – )
**Archibald James Leofric Temple Hamilton-Temple-Blackwood, 2nd Earl of Ava (17 November 1888? – 11 January 1900)
**Terence John Temple Hamilton-Temple-Blackwood, 3rd Earl of Ava (11 January 1900 – 12 February 1902)&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|date=2020-12-04|title=Terence Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Terence_Hamilton-Temple-Blackwood,_2nd_Marquess_of_Dufferin_and_Ava&amp;oldid=992359425|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Terence_Hamilton-Temple-Blackwood,_2nd_Marquess_of_Dufferin_and_Ava.&lt;/ref&gt;
**Frederick Temple Hamilton-Temple-Blackwood, 4th Earl of Ava (12 February 1902 – 21 July 1930)

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1893 October 16''', Lady Florence Davis and Terence John Temple Hamilton-Temple-Blackwood married.&lt;ref name=&quot;:7&quot;&gt;&quot;Florence Davis.&quot; {{Cite web|url=https://www.thepeerage.com/p1068.htm#i10671|title=Person Page|website=www.thepeerage.com|access-date=2021-05-31}} https://www.thepeerage.com/p1068.htm#i10671.&lt;/ref&gt;

'''1897 July 2, Friday''', Lady Florence and Lord Terence Blackwood attended the Duchess of Devonshire's [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did the Earl of Ava and Lord Basil Blackwood and, possibly, a Lord and Lady J. Blackwood.

'''1902 February 12''', Terence John Temple Hamilton-Temple-Blackwood succeeded as 2nd Marquess of Dufferin and Ava.&lt;ref name=&quot;:6&quot; /&gt;

'''1919 December 11''', Florence Davis Hamilton-Temple-Blackwood and Richard George Penn Curzon, 4th [[Social Victorians/People/Howe | Earl Howe]] married.&lt;ref name=&quot;:7&quot; /&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
At the Duchess of Devonshire’s 1897 [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lady Florence Blackwood (at 637) was dressed as Flora, Goddess of Flowers. She attended with her husband Lord Terence Blackwood (at 638) and his brother Archibald, Earl of Ava (at 357).

=== Lady Florence Blackwood ===
[[File:Florence-ne-Davis-Marchioness-of-Dufferin-and-Ava-later-Countess-Howe-when-Lady-Terence-Blackwood-as-Flora-Goddess-of-Flowers.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume decorated all over with flowers|Florence, Lady Blackwood as Flora Goddess of Flowers. ©National Portrait Gallery, London.]][[File:Harmensz van Rijn Rembrandt - Флора - Google Art Project.jpg|thumb|Saskia as Flora, Rembrandt]]
Florence, Lady Blackwood called herself Flora.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2021-03-08|title=Flora Curzon, Lady Howe|url=https://en.wikipedia.org/w/index.php?title=Flora_Curzon,_Lady_Howe&amp;oldid=1011048798|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Flora_Curzon,_Lady_Howe.&lt;/ref&gt;

Gunn &amp; Stuart's portrait of &quot;Florence (née Davis), Marchioness of Dufferin and Ava (later Countess Howe) when Lady Terence Blackwood as Flora Goddess of Flowers&quot; in costume is photogravure #41 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:2&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Terence Blackwood, as Flora Goddess of Flowers,&quot; with a Long S in ''Goddess''.&lt;ref&gt;&quot;Florence (née Davis), Marchioness of Dufferin and Ava (later Countess Howe) when Lady Terence Blackwood as Flora Goddess of Flowers.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158393/Florence-ne-Davis-Marchioness-of-Dufferin-and-Ava-later-Countess-Howe-when-Lady-Terence-Blackwood-as-Flora-Goddess-of-Flowers.&lt;/ref&gt;

What information is available about Lady Florence Blackwood's costume does not name an original for it, if one exists. Flora was the Roman goddess of flowers, springtime, and fertility. Rembrandt's 1634 portrait of ''Saskia as Flora''&lt;ref&gt;{{Citation|title=Flora title QS:P1476,en:&quot;Flora&quot;|url=https://commons.wikimedia.org/wiki/File:Harmensz_van_Rijn_Rembrandt_-_%D0%A4%D0%BB%D0%BE%D1%80%D0%B0_-_Google_Art_Project.jpg|date=1634|accessdate=2022-01-31|last=Rembrandt}}. https://commons.wikimedia.org/wiki/File:Harmensz_van_Rijn_Rembrandt_-_Флора_-_Google_Art_Project.jpg.&lt;/ref&gt; (right) suggests some elements common to this goddess, a floral headdress and the staff decorated with flowers.

[[File:Terence-John-Temple-Hamilton-Temple-Blackwood-2nd-Marquess-of-Dufferin-and-Ava-when-Lord-Terence-Blackwood-as-Captain-Blackwood-RN.jpg|thumb|alt=Black-and-white photograph of a standing woman in an historical naval uniform|Lord Terence Blackwood as Captain Blackwood, R.N. ©National Portrait Gallery, London.]]
=== Lord Terence Blackwood ===
Lord Terence Blackwood (at 638) was dressed as Captain Blackwood, Royal Navy.

Gunn &amp; Stuart's portrait of &quot;Terence John Temple Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava when Lord Terence Blackwood as Captain Blackwood, R.N.&quot; in costume is photogravure #42 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:2&quot; /&gt; The printing on the portrait says, &quot;Lord Terence Blackwood as Captain Blackwood, R.N.&quot;&lt;ref&gt;&quot;Terence John Temple Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava when Lord Terence Blackwood as Captain Blackwood, R.N..&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158395/Terence-John-Temple-Hamilton-Temple-Blackwood-2nd-Marquess-of-Dufferin-and-Ava-when-Lord-Terence-Blackwood-as-Captain-Blackwood-RN.&lt;/ref&gt;

Lord Terence Blackwood had a couple of ancestors who were in the Royal Navy, Sir Henry Blackwood (1770–1832), who was a Vice-Admiral&lt;ref&gt;{{Cite journal|date=2021-11-30|title=Henry Blackwood|url=https://en.wikipedia.org/w/index.php?title=Henry_Blackwood&amp;oldid=1057973605|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Henry_Blackwood.&lt;/ref&gt;; Captain Francis Price Blackwood (1809–1854), known for his influence in sailing, especially around Australia&lt;ref&gt;{{Cite journal|date=2020-06-03|title=Francis Price Blackwood|url=https://en.wikipedia.org/w/index.php?title=Francis_Price_Blackwood&amp;oldid=960526518|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Francis_Price_Blackwood.&lt;/ref&gt;; Price Blackwood, 4th Baron Dufferin and Claneboye (1794–1841), also a captain in the Royal Navy.&lt;ref&gt;{{Cite journal|date=2020-06-03|title=Francis Price Blackwood|url=https://en.wikipedia.org/w/index.php?title=Francis_Price_Blackwood&amp;oldid=960526518|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Francis_Price_Blackwood.&lt;/ref&gt; Technically, if Lord Blackwood followed the Duchess of Devonshire's intructions for people to come dressed as someone before 1820 literally, all these Blackwoods are too late, except perhaps Sir Henry Blackwood in an early moment in his life.

=== Archibald Hamilton-Temple-Blackwood, Earl of Ava ===
Archibald Hamilton-Temple-Blackwood, Earl of Ava (at 357) was present.&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

* He was dressed as Archduke Maximilian in the Empress Maria Theresa Quadrille.&lt;ref&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; (p. ??, Col. 1b) &lt;ref&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}}
* He is represented in the article in the ''Gentlewoman'' with a line drawing and this description of a &quot;black and blue satin costume.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 2a}} He is #14 in a group of men, bottom row, second from left, and identified as &quot;Lord Ava (Louis XV.).&quot;
If the ''Times'' and the ''Morning Post'' are correct, he was dressed as Archduke Maximilian Francis of Austria (1756–1801), son of Empress Maria Thérèse and brother of Marie Antoinette.&lt;ref&gt;{{Cite journal|date=2022-01-01|title=Archduke Maximilian Francis of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduke_Maximilian_Francis_of_Austria&amp;oldid=1063099511|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Archduke_Maximilian_Francis_of_Austria.&lt;/ref&gt;

Henry Arthur Cadogan, [[Social Victorians/People/Cadogan|Viscount Chelsea]] was dressed as Le Roi Louis XV and danced in the Louis XV and Louis XVI quadrille with Georgiana, [[Social Victorians/People/Howe|Viscountess Curzon]], as La Reine Marie Leszuiska as his wife, Queen of France. The Earl of Ava could have been a second Louis XV, of course, if the ''Gentlewoman'' is correct.

=== Lord Basil Blackwood ===
Lord Basil Blackwood — Lord Basil (Ian Basil Gawaine Temple) Hamilton-Temple-Blackwood — (at 582) also attended.&lt;ref name=&quot;:4&quot; /&gt;

=== Lord and Lady J. Blackwood ===
Lord J. Blackwood (at 698) and Lady J. Blackwood (at 699) attended the ball.&lt;ref name=&quot;:4&quot; /&gt;

== Demographics ==
*Nationality
**The Hamilton-Temple-Blackwoods: Anglo-Irish&lt;ref name=&quot;:0&quot; /&gt;
**Flora David: American&lt;ref name=&quot;:1&quot; /&gt;
== Family ==
*Frederick Temple Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava (21 June 1826 – 12 February 1902)&lt;ref name=&quot;:0&quot; /&gt;
*Hariot Georgina Rowan-Hamilton (5 February 1843 – 25 October 1936)&lt;ref&gt;&quot;Hariot Georgina Rowan-Hamilton.&quot; {{Cite web|url=https://www.thepeerage.com/p4999.htm#i49985|title=Person Page|website=www.thepeerage.com|access-date=2021-05-31}} https://www.thepeerage.com/p4999.htm#i49985.&lt;/ref&gt;
*# Lady Helen Hermione Hamilton-Temple-Blackwood (1863 – 9 April 1941)
*# Archibald James Leofric Temple Hamilton-Temple-Blackwood, Earl of Ava (28 July 1863 – 11 January 1900)
*# '''Terence John Temple Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava''' (16 March 1866 – 11 February 1919)
*#Sydney Temple Blackwood (29 May 1867 – 29 May 1867)&lt;ref name=&quot;:3&quot; /&gt;
*# Lady Hermione Catherine Helen Hamilton-Temple-Blackwood (1869 – 19 October 1960)
*# Lord Basil&lt;ref&gt;&quot;Lord Ian Basil Gawaine Temple Hamilton-Temple-Blackwood.&quot; {{Cite web|url=https://www.thepeerage.com/p4999.htm#i49988|title=Person Page|website=www.thepeerage.com|access-date=2021-06-01}} https://www.thepeerage.com/p4999.htm#i49988.&lt;/ref&gt; (Ian Basil Gawaine Temple) Hamilton-Temple-Blackwood (4 November 1870 – 3 July 1917)
*# Lady Victoria Alexandrina Hamilton-Temple-Blackwood (1873 – 11 February 1968)
*# Frederick Temple Hamilton-Temple-Blackwood, 3rd Marquess of Dufferin and Ava (26 February 1875 – 21 July 1930)


*Flora (Florence) Hamilton Davis (c. 1865&lt;ref name=&quot;:1&quot; /&gt; – 14 April 1925)&lt;ref name=&quot;:7&quot; /&gt;
#Lady Doris Gwendoline Blackwood (14 December 1895 – 1984&lt;ref name=&quot;:1&quot; /&gt;)
#Lady Ursula Florence Blackwood (9 February 1899 – 1982&lt;ref name=&quot;:1&quot; /&gt;)
#Lady Patricia Ethel Blackwood (20 March 1902 – 1983&lt;ref name=&quot;:1&quot; /&gt;)
*Terence John Temple Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava (16 March 1866 – 11 February 1919)&lt;ref&gt;&quot;Terence John Temple Hamilton-Temple-Blackwood, 2nd Marquess of Dufferin and Ava.&quot; {{Cite web|url=https://www.thepeerage.com/p4999.htm#i49983|title=Person Page|website=www.thepeerage.com|access-date=2021-05-31}} https://www.thepeerage.com/p4999.htm#i49983.&lt;/ref&gt;
*Richard George Penn Curzon, 4th [[Social Victorians/People/Howe | Earl Howe]] (28 April 1861 – 10 January 1929)&lt;ref&gt;&quot;Richard George Penn Curzon, 4th Earl Howe.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106325|title=Person Page|website=www.thepeerage.com|access-date=2021-05-31}} https://www.thepeerage.com/p10633.htm#i106325.&lt;/ref&gt;

== Biographical Material ==

# The Irish Archives Resource has materials on this family:&lt;ref&gt;{{Cite web|url=https://iar.ie/archive/dufferin-ava-papers/|title=Dufferin and Ava Papers|website=Irish Archives Resource|language=en-US|access-date=2024-05-02}} https://iar.ie/archive/dufferin-ava-papers/.&lt;/ref&gt;
#* Frederick Temple Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava: D1071/H Papers of Frederick, 1st Marquess of Dufferin
#* Hariot Georgina Rowan-Hamilton, D1071/J Papers of Hariot, Lady Dufferin
#* Their children: D1071/K Letters of the Blackwood children

== Notes and Questions ==

# It seems unlikely that Terence John Temple Hamilton-Temple-Blackwood would have been called ''Lord J. Blackwood''; he seems to have gone by ''Terence''. I see no others whose surname was Blackwood and whose name would begin with J. Perhaps the newspaper reporter in the ''Times'' meant to write Lord T. Blackwood? In cursive, perhaps the letters were not clear?

== Footnotes ==
{{reflist}}</text>
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  <page>
    <title>Social Victorians/People/Curzon</title>
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      <text bytes="11108" xml:space="preserve">== Also Known As ==
*Family name: Curzon and Curzon-Howe
*Viscount Curzon, courtesy title for the eldest son and presumptive heir of the [[Social Victorians/People/Howe | Earl of Howe]].
*Baron Scarsdale
**Alfred Nathaniel Holden Curzon, 4th Baron Scarsdale (12 November 1856 – 23 March 1916)&lt;ref name=&quot;:2&quot;&gt;&quot;Alfred Nathaniel Holden Curzon, 4th Baron Scarsdale.&quot; {{Cite web|url=https://www.thepeerage.com/p5349.htm#i53482|title=Person Page|website=www.thepeerage.com|access-date=2020-11-21}} https://www.thepeerage.com/p5349.htm#i53482.&lt;/ref&gt;
**George Nathaniel Curzon, 5th Baron Scarsdale (23 March 1916 – 20 March 1925)&lt;ref name=&quot;:1&quot;&gt;&quot;George Nathaniel Curzon, 1st and Last Marquess Curzon of Kedleston.&quot; {{Cite web|url=https://www.thepeerage.com/p5356.htm#i53558|title=Person Page|website=www.thepeerage.com|access-date=2020-11-21}} https://www.thepeerage.com/p5356.htm#i53558.&lt;/ref&gt;
*Baron Curzon of Kedleston (Ireland peerage, created 11 November 1898)
**George Nathaniel Curzon, 1st Baron Curzon of Kedleston (11 November 1898 – 20 March 1925)&lt;ref name=&quot;:1&quot; /&gt;
*Viscount Scarsdale (remainder to George Curzon's father)&lt;ref name=&quot;:1&quot; /&gt;
**George Nathaniel Curzon, 1st Viscount Scarsdale (2 November 1911 – 20 March 1925)&lt;ref name=&quot;:1&quot; /&gt;
*Earl Curzon of Kedleston
**George Nathaniel Curzon, 1st Earl Curzon of Kedleston (2 November 1911 – 20 March 1925)&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|date=2020-11-12|title=George Curzon, 1st Marquess Curzon of Kedleston|url=https://en.wikipedia.org/w/index.php?title=George_Curzon,_1st_Marquess_Curzon_of_Kedleston&amp;oldid=988354081|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/George_Curzon,_1st_Marquess_Curzon_of_Kedleston.&lt;/ref&gt;
*Baron Ravensdale (remainder to George Curzon's daughters)&lt;ref name=&quot;:1&quot; /&gt;
**George Nathaniel Curzon, 1st Baron Ravensdale (2 November 1911 – 20 March 1925)&lt;ref name=&quot;:1&quot; /&gt;
*Marquess Curzon of Kedleston
**George Nathaniel Curzon, 1st Marquess Curzon of Kedleston (28 June 1912 – 20 March 1925)&lt;ref name=&quot;:1&quot; /&gt;

== Acquaintances, Friends and Enemies ==

=== Hon. George Curzon ===
*Romantic relationship: Elinor Glyn, who found out he was engaged to his second wife by reading it in the newspapers, while she was visiting him at his house&lt;ref name=&quot;:3&quot; /&gt;

== Organizations ==

=== Hon. George Curzon&lt;ref name=&quot;:1&quot; /&gt; ===
*Eton College
*Balliol College, Oxford University
*Member,[[Social Victorians/People/The Souls | The Souls]]
*Assistant Private Secretary to 3rd Marquess of Salisbury (1885)
*Member of Parliament for Lancashire, Southport Division (1886–1889)
*Under-Secretary of State for India (1891–1892)
*Privy Counsellor (P.C.) (1895)
*Under-Secretary of State for Foreign Affairs (1895–1898)
*Viceroy of India (January 1899 – 1905)

== Timeline ==
'''1895 April 22''', George Nathaniel Curzon and Mary Victoria Leiter married.&lt;ref name=&quot;:4&quot;&gt;&quot;Mary Victoria Leiter.&quot; {{Cite web|url=https://www.thepeerage.com/p14314.htm#i143134|title=Person Page|website=www.thepeerage.com|access-date=2020-11-21}} https://www.thepeerage.com/p14314.htm#i143134.&lt;/ref&gt;

'''1897 July 2, Friday''', The Hon. George Curzon and the Hon. Mrs. George Curzon attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. (Mary Curzon was #301 in the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; George Curzon was #495.)

'''1902 May 1''', Grace Elvina Hinds and Alfred Hubert Duggan married.&lt;ref name=&quot;:5&quot;&gt;&quot;Grace Elvina Hinds.&quot; {{Cite web|url=https://www.thepeerage.com/p27943.htm#i279426|title=Person Page|website=www.thepeerage.com|access-date=2021-12-05}} https://www.thepeerage.com/p27943.htm#i279426.&lt;/ref&gt;

'''1917 January 2''', George Nathaniel Curzon and Grace Elvina Hinds married.&lt;ref name=&quot;:5&quot; /&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== Hon. Mrs. Mary Curzon ===
[[File:Mary-Victoria-ne-Leiter-Lady-Curzon-of-Kedleston-as-Valentina-Visconti-of-Milan-AD-1447.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a large hat and a long train|The Hon. Mary Curzon as Valentina Visconti of Milan, A.D. 1447. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], the Hon. Mary Curzon was dressed as Marchesa Malaspina in the Venetians procession&lt;ref&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt; or as Valentina Visconti of Milan, &lt;small&gt;A.D&lt;/small&gt;. 1447. The ''Westminster Gazette'' says, &quot;In the same quadrille [as [[Social Victorians/People/Arthur Stanley Wilson|Enid Wilson]]] was the Hon. Mrs. George Curzon, a perfect picture in white and gold brocade, silver sleeves, and a big blue velvet hat.&quot;&lt;ref&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|p. 5, Col. 1}}

Alice Hughes's portrait (left) of &quot;Mary Victoria (née Leiter), Lady Curzon of Kedleston as Valentina Visconti of Milan, A.D. 1447&quot; in costume is photogravure #190 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Mrs George Curzon as Valentina Visconti of Milan, A.D. 1447.&quot;&lt;ref&gt;&quot;Hon. Mrs George Curzon as Valentina Visconti.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158553/Mary-Victoria-ne-Leiter-Lady-Curzon-of-Kedleston-as-Valentina-Visconti-of-Milan-AD-1447.&lt;/ref&gt;

[[Social Victorians/People/Salisbury#Lord Robert Cecil and Lady Eleanor Cecil|Lady Robert Cecil]] went as Valentia Visconti, probably Valentina Visconti, according to her Hughes's portrait in costume. A 15th-century Lady Valentia Visconti is difficult to identify. Two notable Valentina Viscontis lived at the end of the 14th century: Valentina Visconti, Queen of Cyprus (c. 1357 – before September 1393),&lt;ref&gt;{{Cite journal|date=2021-07-01|title=Valentina Visconti, Queen of Cyprus|url=https://en.wikipedia.org/w/index.php?title=Valentina_Visconti,_Queen_of_Cyprus&amp;oldid=1031354412|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Valentina_Visconti,_Queen_of_Cyprus.&lt;/ref&gt; and Valentina Visconti, Duchess of Orléans (1371 – 4 December 1408).&lt;ref&gt;{{Cite journal|date=2021-09-17|title=Valentina Visconti, Duchess of Orléans|url=https://en.wikipedia.org/w/index.php?title=Valentina_Visconti,_Duchess_of_Orl%C3%A9ans&amp;oldid=1044888748|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Valentina_Visconti,_Duchess_of_Orléans.&lt;/ref&gt; Both were born in Milan.

[[File:Richard George Penn Curzon, Vanity Fair, 1896-06-04.jpg|thumb|''South Bucks'' — George Curzon — by &quot;Spy,&quot; ''Vanity Fair'', 4 June 1896]]
=== Hon. George Curzon ===
The Hon. George Curzon was dressed as a &quot;Spanish Admiral&quot; in &quot;black velvet.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 2a–b}}

No photograph of him in costume exists, but the caricature of him (right) shows him in 1896 as one of ''Vanity Fair''&lt;nowiki/&gt;'s &quot;Statesmen&quot; series, Number 672.&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt;

== Demographics ==

=== Nationality ===
*Mary Victoria Leiter Curzon, American&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-11-16|title=Mary Curzon, Baroness Curzon of Kedleston|url=https://en.wikipedia.org/w/index.php?title=Mary_Curzon,_Baroness_Curzon_of_Kedleston&amp;oldid=989054984|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Grace Elvina Hinds, American&lt;ref name=&quot;:5&quot; /&gt;

== Family ==
*Alfred Nathaniel Holden Curzon, 4th Baron Scarsdale (12 July 1831 – 23 March 1916)&lt;ref name=&quot;:2&quot; /&gt;
*Blanche Senhouse ( – 4 April 1875)&lt;ref&gt;&quot;Blanche Senhouse.&quot; {{Cite web|url=https://www.thepeerage.com/p7216.htm#i72159|title=Person Page|website=www.thepeerage.com|access-date=2020-11-21}} https://www.thepeerage.com/p7216.htm#i72159.&lt;/ref&gt;
#Hon. Sophia Caroline Curzon (20 Nov 1857 – 12 March 1929)
#'''George Nathaniel Curzon''', 1st and last Marquess Curzon of Kedleston (11 January 1859 – 20 March 1925)
#Colonel Hon. Alfred Nathaniel Curzon (12 March 1860 – 20 September 1920)
#Hon. Mary Eveline Curzon (c. 1861 – 4 May 1862)
#Hon. Blanche Felicia Curzon (18 April 1861 – 23 April 1928)
#Hon. Eveline Mary Curzon (16 February 1864 – 15 June 1934)
#Hon. Francis Nathaniel Curzon (15 December 1865 – 8 June 1941)
#Hon. Assheton Nathaniel Curzon (10 May 1867 – 20 August 1950)
#Hon. Elinor Florence Curzon (13 February 1869 – 11 March 1939)
#Hon. Geraline Emily Curzon (8 May 1871 – 17 May 1940)
#Hon. Margaret Georgina Curzon (23 December 1873 – 19 July 1957)


*George Nathaniel Curzon, 1st Marquess Curzon of Kedleston (11 January 1859 – 20 March 1925)&lt;ref name=&quot;:1&quot; /&gt;
*Mary Victoria Leiter ( – 18 July 1906)&lt;ref name=&quot;:4&quot; /&gt;
#Mary Irene Curzon, Baroness Ravensdale of Kedleston (20 January 1896 – 9 February 1966)
#Cynthia Blanche Curzon (23 August 1898 – 16 May 1933)
#Alexandra Naldera Curzon (20 March 1904 – 7 August 1995)

* Grace Elvina Hinds ( – 29 June 1958)&lt;ref name=&quot;:5&quot; /&gt;

=== Relations ===
*Daisy (Marguerite Hyde)[[Social Victorians/People/Suffolk | Leiter Howard]] and Mary Victoria Leiter Curzon were sisters.&lt;ref name=&quot;:0&quot; /&gt;

== Biographical Material ==

# Looked in the Irish Archives Resource for ''Curzon'' and ''Kedleston'', turning up nothing relevant.

== Notes and Questions ==
A number of people named Curzon attended the Duchess of Devonshire's ball:
#Viscount and Viscountess Curzon are treated on the page for the [[Social Victorians/People/Howe | Earl of Howe]].
#The ''Times'' article&lt;ref&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; lists both Mrs. George Curzon and then later Mr. and Mrs. Curzon; was this another couple? Several people are treated this way, mentioned earlier in the ''Times'' article and then apparently showing up later, with fewer honorifics, especially if they are Hon.'s.

== Footnotes ==
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      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10076" xml:space="preserve">[[File:Lord Dungarvan Vanity Fair 1897-10-28.jpg|thumb|alt=Colored drawing of a standing man in a 19th-century suit, black tie and top hat, his left hand in his trouser pocket, his hidden right hand holding a walking stick, facing to his right|''Sol'' — Charles Spencer Canning Boyle, Lord Dungarvan — by &quot;Spy,&quot; ''Vanity Fair'' 28 October 1897]]
==Also Known As==
* Family name: Boyle
* Earl of Cork and Orrery
** Richard Edmund St. Lawrence Boyle, 9th Earl of Cork ( – 22 June 1904)
** Charles Spencer Canning Boyle, 10th Earl of Cork and Orrery (22 June 1904 – 25 March 1925)
* Viscount Dungarvan (a courtesy title for the eldest son and heir apparent of the Earl of Cork and Orrery)
** Charles Spencer Canning Boyle, Viscount Dungarvan (1861 – 22 June 1904)
* Cork

==Acquaintances, Friends and Enemies==
==Organizations==
===Charles Spencer Canning Boyle===
* [[Social Victorians/19thC Freemasonry|Freemasons]] of Somerset (from 1891)

=== Ginger (William Henry Dudley) Boyle ===

* Cadet, HMS Britannia (15 January 1887)&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2021-06-02|title=William Boyle, 12th Earl of Cork and Orrery|url=https://en.wikipedia.org/w/index.php?title=William_Boyle,_12th_Earl_of_Cork_and_Orrery&amp;oldid=1026543676|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/William_Boyle,_12th_Earl_of_Cork_and_Orrery.&lt;/ref&gt;
* Midshipman (15 June 1889)&lt;ref name=&quot;:0&quot; /&gt;
* Sub-lieutenant (1 July 1894)&lt;ref name=&quot;:0&quot; /&gt;
* Lieutenant (1 October 1895)
* First Lieutenant, China Station, Boxer Rebellion (November 1898)&lt;ref name=&quot;:0&quot; /&gt;
* Commanding officer on a destroyer (28 August 1902)&lt;ref name=&quot;:0&quot; /&gt;
* Executive Officer (February 1904)&lt;ref name=&quot;:0&quot; /&gt;
* Commander (31 December 1906)&lt;ref name=&quot;:0&quot; /&gt;
* Captain (30 June 1913)&lt;ref name=&quot;:0&quot; /&gt;
* British naval attaché, Rome (July 1913)&lt;ref name=&quot;:0&quot; /&gt;
* &quot;worked closely with T. E. Lawrence in support of the Arab Revolt&quot;&lt;ref name=&quot;:0&quot; /&gt;

==Timeline==
'''1897 July 2''', Mr. W. Boyle attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]]. (Mr. W. Boyle is #504 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]].) 

'''1897 July 9''', the Earl and Countess of Cork and Orrery were invited to the [[Social Victorians/1891-07-09 Garden Party | Garden Party at Marlborough House]].

'''1902 July 24''', Ginger (William) Boyle and Lady Florence Cecilia Keppel married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Florence Cecilia Keppel.&quot; {{Cite web|url=https://www.thepeerage.com/p1724.htm#i17240|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p1724.htm#i17240.&lt;/ref&gt;

'''1904 June 22''', Charles Spencer Canning Boyle became Earl of Cork on the death of his father.

'''1918 November 21''', Charles Spencer Canning Boyle and Mrs. Rosalie de Villiers Gray married.

==Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball==
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Mr. W. Boyle — possibly William Henry Dudley Boyle — wore an &quot;Elizabethan costume of violet and silver brocade.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|40, Col. 1a}}

== Demographics ==
* Nationality: Irish&lt;ref&gt;{{Cite journal|date=2021-06-02|title=Edmund Boyle, 8th Earl of Cork|url=https://en.wikipedia.org/w/index.php?title=Edmund_Boyle,_8th_Earl_of_Cork&amp;oldid=1026543344|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edmund_Boyle,_8th_Earl_of_Cork.&lt;/ref&gt;; British&lt;ref name=&quot;:0&quot; /&gt;
*Religion: Catholic&lt;ref name=&quot;:0&quot; /&gt;

==Family==

* General Edmund Boyle, 8th Earl of Cork (21 October 1767 – 29 June 1856)&lt;ref&gt;&quot;General Edmund Boyle, 8th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27722|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27722.&lt;/ref&gt;
* Isabella Henrietta Poyntz ( – 29 November 1843)&lt;ref&gt;&quot;Isabella Henrietta Poyntz.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27723|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27723.&lt;/ref&gt;
*# Edmund William Boyle, Viscount Dungarvan (2 October 1798 – 1 January 1826)
*# '''Charles Boyle, Viscount Dungarvan''' (6 December 1800 – 25 August 1834)
*# '''Hon. John Boyle''' (13 March 1803 – 6 December 1874)
*# Colonel Hon. Robert Edward Boyle (March 1809 – 3 September 1854)
*# Reverend Hon. Richard Cavendish Boyle (28 Feb 1812 – 30 March 1886)


* Charles Boyle, Viscount Dungarvan (6 December 1800 – 25 August 1834)&lt;ref&gt;&quot;Charles Boyle, Viscount Dungarvan.&quot; {{Cite web|url=https://www.thepeerage.com/p2435.htm#i24344|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2435.htm#i24344.&lt;/ref&gt;
* Lady Catherine St. Lawrence ( – 4 April 1879)&lt;ref&gt;&quot;Lady Catherine St. Lawrence.&quot; {{Cite web|url=https://www.thepeerage.com/p2772.htm#i27720|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2772.htm#i27720.&lt;/ref&gt;
*# Louisa Caroline Elizabeth Boyle ( – 5 May 1876)
*# Lady Mary Emily Boyle ( – 25 November 1916)
*# '''Richard Edmund St. Lawrence Boyle, 9th Earl of Cork''' (19 April 1829 – 22 June 1904)
*# Lt.-Col. Hon. William George Boyle (12 August 1830 – 24 March 1908)
*# Major Hon. Edmund John Boyle (25 November 1831 – 21 April 1901)


* Hon. John Boyle (13 March 1803 – 6 December 1874)&lt;ref&gt;&quot;Hon. John Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27734|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27734.&lt;/ref&gt;
* Hon. Cecilia FitzGerald-de Ros ( – 6 October 1869)&lt;ref&gt;&quot;Hon. Cecilia FitzGerald-de Ros.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27735|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27735.&lt;/ref&gt;
*# '''Colonel Gerald Edmund Boyle''' (20 June 1840 – 28 December 1927)
*# Robert Frederick Boyle (13 January 1841 – 15 May 1883)
*# Georgiana Olivia Boyle (27 January 1843 – 15 January 1931)
*# Edmund Montague Boyle (17 July 1845 – 11 August 1885)


* Richard Edmund St. Lawrence Boyle, 9th Earl of Cork (19 April 1829 – 22 June 1904)&lt;ref&gt;&quot;Richard Edmund St. Lawrence Boyle, 9th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p2435.htm#i24345|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2435.htm#i24345.&lt;/ref&gt;
* Lady Emily Charlotte de Burgh (19 October 1828 – 10 October 1912)&lt;ref&gt;&quot;Lady Emily Charlotte de Burgh.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27726|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27726.&lt;/ref&gt;
# Lady Emily Harriet Catherine Boyle (c. 1855 – 28 July 1931)
# Lady Grace Elizabeth Boyle (c. 1858 – 23 May 1935)
# Lady Honora Janet Boyle (c. 1859 – 11 March 1953)
# Lady Isabel Lettice Theodosia Boyle (d. 6 April 1904)
# Lady Dorothy Blanche Boyle (c. 1860 – 7 June 1938)
# '''Charles Spencer Boyle''', 10th Earl of Cork (24 November 1861 – 25 March 1925)
# Robert John Lascelles Boyle, 11th Earl of Cork (1864–1934)


* Colonel Gerald Edmund Boyle (20 June 1840 – 28 December 1927)&lt;ref&gt;&quot;Colonel Gerald Edmund Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27738|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27738.&lt;/ref&gt;
* Lady Elizabeth Theresa Pepys ( – 24 January 1897)&lt;ref&gt;&quot;Lady Elizabeth Theresa Pepys.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27739|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27739.&lt;/ref&gt;
*# Lady Cecilia Georgiana Boyle ( – 16 April 1952)
*# Lady Theresa Selina Boyle ( – 29 June 1956)
*# Major Arthur Gerald Boyle (26 July 1865 – 30 June 1912)
*# Lady Caroline Elizabeth Boyle (c. 1870 – 4 December 1958)
*# Admiral '''William Henry Dudley Boyle, 12th Earl of Cork''' (30 November 1873 – 19 April 1967)
*# Evelyn Blanche Boyle (30 November 1873 – 29 June 1898)
*# Captain Hon. Frederick John Boyle (4 July 1875 – 18 October 1955)
*# Major Hon. Reginald Courtenay Boyle (22 November 1877 – 16 February 1946)

* Charles Spencer Canning Boyle, 10th Earl of Cork and Orrery (24 November 1861 – 25 March 1925)
* Rosalie de Villiers Gray Boyle ( – 15 March 1930)


* Admiral Ginger&lt;ref name=&quot;:0&quot; /&gt; (William Henry Dudley) Boyle, 12th Earl of Cork (30 November 1873 – 19 April 1967)&lt;ref&gt;&quot;Admiral William Henry Dudley Boyle, 12th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p1725.htm#i17241|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p1725.htm#i17241.&lt;/ref&gt;
* Lady Florence Cecilia Keppel (24 February 1871 – 30 June 1963)&lt;ref name=&quot;:1&quot; /&gt;

==Biographical Materials==

# The ''Irish Archives Resource'' doesn't seem to have anything for these people at the end of the 19th century. 

==Questions and Notes==

# While members of this extended family attended notable social events about this time, no other Boyle attended the Duchess of Devonshire's ball. Ginger (William Henry Dudley) Boyle — later Admiral and 12th Earl of Cork — could conceivably be this Mr. W. Boyle, 
#Mr. W. Boyle is probably not Lt.-Col. Hon. William George Boyle, who was granted the rank of an earl's younger son, which makes the honorific &quot;Hon&quot; rather than &quot;Mr.&quot;; he attained his rank as Lieutenant-Colonel in the Coldstream Guards.&lt;ref&gt;&quot;Lt.-Col. Hon. William George Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p1109.htm#i11087|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p1109.htm#i11087.&lt;/ref&gt;

==Footnotes==
{{reflist}}</text>
      <sha1>judfxzkfb544bnd24ibj5910gfbftjo</sha1>
    </revision>
    <revision>
      <id>2624855</id>
      <parentid>2624840</parentid>
      <timestamp>2024-05-02T22:22:54Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10119" xml:space="preserve">[[File:Lord Dungarvan Vanity Fair 1897-10-28.jpg|thumb|alt=Colored drawing of a standing man in a 19th-century suit, black tie and top hat, his left hand in his trouser pocket, his hidden right hand holding a walking stick, facing to his right|''Sol'' — Charles Spencer Canning Boyle, Lord Dungarvan — by &quot;Spy,&quot; ''Vanity Fair'' 28 October 1897]]
==Also Known As==
* Family name: Boyle
* Earl of Cork and Orrery
** Richard Edmund St. Lawrence Boyle, 9th Earl of Cork ( – 22 June 1904)
** Charles Spencer Canning Boyle, 10th Earl of Cork and Orrery (22 June 1904 – 25 March 1925)
* Viscount Dungarvan (a courtesy title for the eldest son and heir apparent of the Earl of Cork and Orrery)
** Charles Spencer Canning Boyle, Viscount Dungarvan (1861 – 22 June 1904)
* Cork

==Acquaintances, Friends and Enemies==
==Organizations==
===Charles Spencer Canning Boyle===
* [[Social Victorians/19thC Freemasonry|Freemasons]] of Somerset (from 1891)

=== Ginger (William Henry Dudley) Boyle ===

* Cadet, HMS Britannia (15 January 1887)&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2021-06-02|title=William Boyle, 12th Earl of Cork and Orrery|url=https://en.wikipedia.org/w/index.php?title=William_Boyle,_12th_Earl_of_Cork_and_Orrery&amp;oldid=1026543676|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/William_Boyle,_12th_Earl_of_Cork_and_Orrery.&lt;/ref&gt;
* Midshipman (15 June 1889)&lt;ref name=&quot;:0&quot; /&gt;
* Sub-lieutenant (1 July 1894)&lt;ref name=&quot;:0&quot; /&gt;
* Lieutenant (1 October 1895)
* First Lieutenant, China Station, Boxer Rebellion (November 1898)&lt;ref name=&quot;:0&quot; /&gt;
* Commanding officer on a destroyer (28 August 1902)&lt;ref name=&quot;:0&quot; /&gt;
* Executive Officer (February 1904)&lt;ref name=&quot;:0&quot; /&gt;
* Commander (31 December 1906)&lt;ref name=&quot;:0&quot; /&gt;
* Captain (30 June 1913)&lt;ref name=&quot;:0&quot; /&gt;
* British naval attaché, Rome (July 1913)&lt;ref name=&quot;:0&quot; /&gt;
* &quot;worked closely with T. E. Lawrence in support of the Arab Revolt&quot;&lt;ref name=&quot;:0&quot; /&gt;

==Timeline==
'''1897 July 2''', Mr. W. Boyle attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]]. (Mr. W. Boyle is #504 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]].) 

'''1897 July 9''', the Earl and Countess of Cork and Orrery were invited to the [[Social Victorians/1891-07-09 Garden Party | Garden Party at Marlborough House]].

'''1902 July 24''', Ginger (William) Boyle and Lady Florence Cecilia Keppel married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Florence Cecilia Keppel.&quot; {{Cite web|url=https://www.thepeerage.com/p1724.htm#i17240|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p1724.htm#i17240.&lt;/ref&gt;

'''1904 June 22''', Charles Spencer Canning Boyle became Earl of Cork on the death of his father.

'''1918 November 21''', Charles Spencer Canning Boyle and Mrs. Rosalie de Villiers Gray married.

==Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball==
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Mr. W. Boyle — possibly William Henry Dudley Boyle — wore an &quot;Elizabethan costume of violet and silver brocade.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|40, Col. 1a}}

== Demographics ==
* Nationality: Irish&lt;ref&gt;{{Cite journal|date=2021-06-02|title=Edmund Boyle, 8th Earl of Cork|url=https://en.wikipedia.org/w/index.php?title=Edmund_Boyle,_8th_Earl_of_Cork&amp;oldid=1026543344|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edmund_Boyle,_8th_Earl_of_Cork.&lt;/ref&gt;; British&lt;ref name=&quot;:0&quot; /&gt;
*Religion: Catholic&lt;ref name=&quot;:0&quot; /&gt;

==Family==

* General Edmund Boyle, 8th Earl of Cork (21 October 1767 – 29 June 1856)&lt;ref&gt;&quot;General Edmund Boyle, 8th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27722|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27722.&lt;/ref&gt;
* Isabella Henrietta Poyntz ( – 29 November 1843)&lt;ref&gt;&quot;Isabella Henrietta Poyntz.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27723|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27723.&lt;/ref&gt;
*# Edmund William Boyle, Viscount Dungarvan (2 October 1798 – 1 January 1826)
*# '''Charles Boyle, Viscount Dungarvan''' (6 December 1800 – 25 August 1834)
*# '''Hon. John Boyle''' (13 March 1803 – 6 December 1874)
*# Colonel Hon. Robert Edward Boyle (March 1809 – 3 September 1854)
*# Reverend Hon. Richard Cavendish Boyle (28 Feb 1812 – 30 March 1886)


* Charles Boyle, Viscount Dungarvan (6 December 1800 – 25 August 1834)&lt;ref&gt;&quot;Charles Boyle, Viscount Dungarvan.&quot; {{Cite web|url=https://www.thepeerage.com/p2435.htm#i24344|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2435.htm#i24344.&lt;/ref&gt;
* Lady Catherine St. Lawrence ( – 4 April 1879)&lt;ref&gt;&quot;Lady Catherine St. Lawrence.&quot; {{Cite web|url=https://www.thepeerage.com/p2772.htm#i27720|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2772.htm#i27720.&lt;/ref&gt;
*# Louisa Caroline Elizabeth Boyle ( – 5 May 1876)
*# Lady Mary Emily Boyle ( – 25 November 1916)
*# '''Richard Edmund St. Lawrence Boyle, 9th Earl of Cork''' (19 April 1829 – 22 June 1904)
*# Lt.-Col. Hon. William George Boyle (12 August 1830 – 24 March 1908)
*# Major Hon. Edmund John Boyle (25 November 1831 – 21 April 1901)


* Hon. John Boyle (13 March 1803 – 6 December 1874)&lt;ref&gt;&quot;Hon. John Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27734|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27734.&lt;/ref&gt;
* Hon. Cecilia FitzGerald-de Ros ( – 6 October 1869)&lt;ref&gt;&quot;Hon. Cecilia FitzGerald-de Ros.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27735|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27735.&lt;/ref&gt;
*# '''Colonel Gerald Edmund Boyle''' (20 June 1840 – 28 December 1927)
*# Robert Frederick Boyle (13 January 1841 – 15 May 1883)
*# Georgiana Olivia Boyle (27 January 1843 – 15 January 1931)
*# Edmund Montague Boyle (17 July 1845 – 11 August 1885)


* Richard Edmund St. Lawrence Boyle, 9th Earl of Cork (19 April 1829 – 22 June 1904)&lt;ref&gt;&quot;Richard Edmund St. Lawrence Boyle, 9th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p2435.htm#i24345|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2435.htm#i24345.&lt;/ref&gt;
* Lady Emily Charlotte de Burgh (19 October 1828 – 10 October 1912)&lt;ref&gt;&quot;Lady Emily Charlotte de Burgh.&quot; {{Cite web|url=https://www.thepeerage.com/p2773.htm#i27726|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p2773.htm#i27726.&lt;/ref&gt;
# Lady Emily Harriet Catherine Boyle (c. 1855 – 28 July 1931)
# Lady Grace Elizabeth Boyle (c. 1858 – 23 May 1935)
# Lady Honora Janet Boyle (c. 1859 – 11 March 1953)
# Lady Isabel Lettice Theodosia Boyle (d. 6 April 1904)
# Lady Dorothy Blanche Boyle (c. 1860 – 7 June 1938)
# '''Charles Spencer Boyle''', 10th Earl of Cork (24 November 1861 – 25 March 1925)
# Robert John Lascelles Boyle, 11th Earl of Cork (1864–1934)


* Colonel Gerald Edmund Boyle (20 June 1840 – 28 December 1927)&lt;ref&gt;&quot;Colonel Gerald Edmund Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27738|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27738.&lt;/ref&gt;
* Lady Elizabeth Theresa Pepys ( – 24 January 1897)&lt;ref&gt;&quot;Lady Elizabeth Theresa Pepys.&quot; {{Cite web|url=https://www.thepeerage.com/p2774.htm#i27739|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p2774.htm#i27739.&lt;/ref&gt;
*# Lady Cecilia Georgiana Boyle ( – 16 April 1952)
*# Lady Theresa Selina Boyle ( – 29 June 1956)
*# Major Arthur Gerald Boyle (26 July 1865 – 30 June 1912)
*# Lady Caroline Elizabeth Boyle (c. 1870 – 4 December 1958)
*# Admiral '''William Henry Dudley Boyle, 12th Earl of Cork''' (30 November 1873 – 19 April 1967)
*# Evelyn Blanche Boyle (30 November 1873 – 29 June 1898)
*# Captain Hon. Frederick John Boyle (4 July 1875 – 18 October 1955)
*# Major Hon. Reginald Courtenay Boyle (22 November 1877 – 16 February 1946)

* Charles Spencer Canning Boyle, 10th Earl of Cork and Orrery (24 November 1861 – 25 March 1925)
* Rosalie de Villiers Gray Boyle ( – 15 March 1930)


* Admiral Ginger&lt;ref name=&quot;:0&quot; /&gt; (William Henry Dudley) Boyle, 12th Earl of Cork (30 November 1873 – 19 April 1967)&lt;ref&gt;&quot;Admiral William Henry Dudley Boyle, 12th Earl of Cork.&quot; {{Cite web|url=https://www.thepeerage.com/p1725.htm#i17241|title=Person Page|website=www.thepeerage.com|access-date=2021-06-19}} https://www.thepeerage.com/p1725.htm#i17241.&lt;/ref&gt;
* Lady Florence Cecilia Keppel (24 February 1871 – 30 June 1963)&lt;ref name=&quot;:1&quot; /&gt;

==Biographical Materials==

# Looked for Boyle, Orrery and Cork in the ''Irish Archives Resource'', which doesn't seem to have anything for these people at the end of the 19th century.

==Questions and Notes==

# While members of this extended family attended notable social events about this time, no other Boyle attended the Duchess of Devonshire's ball. Ginger (William Henry Dudley) Boyle — later Admiral and 12th Earl of Cork — could conceivably be this Mr. W. Boyle, 
#Mr. W. Boyle is probably not Lt.-Col. Hon. William George Boyle, who was granted the rank of an earl's younger son, which makes the honorific &quot;Hon&quot; rather than &quot;Mr.&quot;; he attained his rank as Lieutenant-Colonel in the Coldstream Guards.&lt;ref&gt;&quot;Lt.-Col. Hon. William George Boyle.&quot; {{Cite web|url=https://www.thepeerage.com/p1109.htm#i11087|title=Person Page|website=www.thepeerage.com|access-date=2021-06-18}} https://www.thepeerage.com/p1109.htm#i11087.&lt;/ref&gt;

==Footnotes==
{{reflist}}</text>
      <sha1>a4x52m6y3j4d4un1o6va6r2jl9p7yvw</sha1>
    </revision>
  </page>
  <page>
    <title>Social Victorians/People/Castlerosse</title>
    <ns>0</ns>
    <id>265028</id>
    <revision>
      <id>2624838</id>
      <parentid>2182501</parentid>
      <timestamp>2024-05-02T20:24:51Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1801" xml:space="preserve">==Also Known As==
* Family name: Browne
* Earl of Kenmare
** Valentine Augustus Browne, 4th Earl of Kenmare (1871? – 9 February 1905)
** Valentine Charles Browne, 5th Earl of Kenmare (1860 – 14 November 1941)
* Viscount Castlerosse was a courtesy title for the eldest son of the Earl of Kenmare in the peerage of Ireland.
* Viscount Castlerosse
** Valentine Augustus Browne, Viscount Castlerosse (1853–1871)
** Valentine Charles Browne, Viscount Castlerosse (1871–1905)

==Acquaintances, Friends and Enemies==
==Timeline==
'''1858 April 28''', Valentine Augustus Browne and Gertrude Thynne married.

== Demographics ==
* Nationality:

==Family==
* Valentine Augustus Browne, 4th Earl of Kenmare (16 May 1825 – 9 February 1905)
* Gertrude Thynne Browne (– Feb. 1913)
# Margaret Theodora May Catherine Browne (d. 1940)
# '''Valentine Charles Browne''', 5th Earl of Kenmare (1860–1941)
# Cecil Augustine Browne (1864–1887)


* Valentine Charles Browne, 5th Earl of Kenmare (1 December 1860 – 14 November 1941)
* He must have married, had at least one daughter
# Dorothy Margaret Browne

== Biographical Material ==

# Kenmare Papers: at PRONI, found via the ''Irish Archives Resource'', 1587-1973, Archive Reference GB 0255 PRONI/D4151.&lt;ref&gt;{{Cite web|url=https://iar.ie/archive/kenmare-papers/|title=Kenmare Papers|website=Irish Archives Resource|language=en-US|access-date=2024-05-02}} https://iar.ie/archive/kenmare-papers/.&lt;/ref&gt; Includes correspondence.

==Questions and Notes==
==Bibliography==
* &quot;Valentine Browne, 4th Earl of Kenmare.&quot; Wikipedia https://en.wikipedia.org/wiki/Valentine_Browne,_4th_Earl_of_Kenmare (accessed June 2015).
* &quot;Valentine Browne, 5th Earl of Kenmare.&quot; Wikipedia https://en.wikipedia.org/wiki/Valentine_Browne,_5th_Earl_of_Kenmare (accessed June 2015).</text>
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{{InformationQ|Q81333757}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
[[Category:WikiJournal]]</text>
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      <model>wikitext</model>
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      <text bytes="303" xml:space="preserve">== Summary ==
{{Information
|Description= Peer review of [[WikiJournal_Preprints/Melioidosis]] with replies from corresponding author
|Source= Raymond Chieng
|Date=13 February 2021
|Author=Raymond Chieng
|Permission=
}}
== Licensing ==
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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{{Information
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|P1476|fetchwikidata=ALL|onlysourced=no|noicon=true}}''
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== Licensing ==
{{self|GFDL|cc-by-4.0}}
&lt;references group=&quot;lower-alpha&quot; /&gt;
[[Category:WikiJournal]]</text>
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      <comment>File have the text WikiJournal somewhere so putting in [[Category:WikiJournal]].</comment>
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      <text bytes="103" xml:space="preserve">== Summary ==
{{InformationQ | Q = Q86161728}}

== License ==
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      <text bytes="376" xml:space="preserve">== Summary ==
{{Information
|Description=Peer review of [[WikiJournal Preprints/“It’s all about people skills”: Perspectives on the social license of the forest products industry from rural North America]]
|Source= Ian Thomson
|Date= 26 February 2021
|Author=Ian Thomson
|Permission=
}}
== Licensing ==
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[[Category:WikiJournal]]</text>
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    <title>Template:Regular convex 4-polytopes</title>
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      <text bytes="14608" xml:space="preserve">{| class=&quot;wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}&quot; style=&quot;white-space:nowrap;text-align:center;&quot;
!colspan=9|[[{{{wiki|}}}Regular 4-polytopes|Regular convex 4-polytopes]] {{#if:{{{radius|}}}|of radius {{{radius|}}}|}}
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Coxeter_group|Symmetry group]]
|[[{{{wiki|}}}Tetrahedral symmetry|A&lt;sub&gt;4&lt;/sub&gt;]]
|colspan=2|[[{{{wiki|}}}Hyperoctahedral_group|B&lt;sub&gt;4&lt;/sub&gt;]]
|[[{{{wiki|}}}F4_(mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]]
|colspan=3|[[{{{wiki|}}}H4_polytope|H&lt;sub&gt;4&lt;/sub&gt;]]
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Name
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}5-cell|5-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}tetrahedron|tetrahedron]]&lt;BR&gt;
5-point
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}16-cell|16-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}octahedron|octahedron]]&lt;BR&gt;
8-point
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}8-cell|8-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}cube|cube]]&lt;BR&gt;
16-point
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}24-cell|24-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Jessen's icosahedron|cuboctahedron]]&lt;BR&gt;
24-point
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}600-cell|600-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Regular icosahedron|icosahedron]]&lt;BR&gt;
120-point
|style=&quot;vertical-align:top;&quot;|[[#One fibration of 11-cells|137-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Rhomibicosadodecahedron|rhomibicosadodecahedron]]&lt;BR&gt;
137-point
|style=&quot;vertical-align:top;&quot;|[[{{{wiki|}}}120-cell|120-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Regular dodecahedron|dodecahedron]]&lt;BR&gt;
600-point
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Schläfli symbol|Schläfli symbol]]
|{3, 3, 3}
|{3, 3, 4}
|{4, 3, 3}
|{3, 4, 3}
|{3, 3, 5}
|{3, 5, 3}
|{5, 3, 3}
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Coxeter diagram|Coxeter mirrors]]
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}}
|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|5|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}
|-
!style=&quot;text-align:right;&quot;|Mirror dihedrals
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|5}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|5}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|5}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Graph
|[[Image:4-simplex t0.svg|120px]]
|[[Image:4-cube t3.svg|120px]]
|[[Image:4-cube t0.svg|120px]]
|[[Image:24-cell t0 F4.svg|120px]]
|[[Image:600-cell graph H4.svg|120px]]
|
|[[Image:120-cell graph H4.svg|120px]]
|-
!style=&quot;text-align:right;&quot;|Vertices{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[W:120-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 600-point (120-cell) 4-polytope is sixth and last in the ascending sequence that begins with the 5-point (5-cell) 4-polytope.|name=4-polytopes ordered by size and complexity}}
|5 tetrahedral
|8 octahedral
|16 tetrahedral
|24 cubical
|120 icosahedral
|137 cuboctahedral
|600 tetrahedral
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}120-cell#Chords|Edges]]
|10 triangular
|24 square
|32 triangular
|96 triangular
|720 pentagonal
|triangular
|1200 triangular
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Faces
|10 triangles
|32 triangles
|24 squares
|96 triangles
|1200 triangles
|triangles
|720 pentagons
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Cells
|5 tetrahedra
|16 tetrahedra
|8 cubes
|24 octahedra
|600 tetrahedra
|137 rhomibicosadodecahedra
|120 dodecahedra
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}600-cell#Clifford parallel cell rings|Tori]]
|1 [[{{{wiki|}}}5-cell#Boerdijk–Coxeter helix|5-tetrahedron]]
|2 [[{{{wiki|}}}16-cell#Helical construction|8-tetrahedron]]
|2 [[{{{wiki|}}}8-cell#Construction|4-cube]]
|4 [[{{{wiki|}}}24-cell#Cell rings|6-octahedron]]
|20 [[{{{wiki|}}}600-cell#Boerdijk–Coxeter helix rings|30-tetrahedron]]
|11 [[{{{wiki|}}}#The 11-point 11-cell|11-rhomibicosadodecahedron]]
|12 [[{{{wiki|}}}120-cell#Intertwining rings|10-dodecahedron]]
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Inscribed
|120 in 120-cell
|675 in 120-cell
|2 16-cells
|3 8-cells
|25 24-cells
|9 600-cells
|11 137-cells
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Pentads
|1
|
|
|
|
|55
|120
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Hexads
|
|1
|2
|3
|75
|600
|675
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Heptads
|
|
|
|
|
|55
|120
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}Great circle|Great polygons]]
|
|2 [[{{{wiki|}}}16-cell#Coordinates|squares]] x 3
|4 rectangles x 4
|4 [[{{{wiki|}}}24-cell#Hexagons|hexagons]] x 4
|12 [[{{{wiki|}}}600-cell#Geodesics|decagons]] x 6
|11 [[{{{wiki|}}}#The rings of the 11-cells|hendecagons]] x 11
|100 [[{{{wiki|}}}120-cell#Chords|irregular hexagons]] x 4
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}Petrie polygon|Petrie polygon]]s
|1 [[{{{wiki|}}}5-cell#Boerdijk–Coxeter helix|pentagon]] x 2
|1 [[{{{wiki|}}}16-cell#Helical construction|octagon]] x 3
|2 [[{{{wiki|}}}Octagon#Skew octagon|octagon]]s x 4
|2 [[{{{wiki|}}}Dodecagon#Skew dodecagon|dodecagon]]s x 4
|4 [[{{{wiki|}}}30-gon#Petrie polygons|30-gon]]s x 6
|11 [[{{{wiki|}}}30-gon#Petrie polygons|30-gon]]s x 11
|20 [[{{{wiki|}}}30-gon#Petrie polygons|30-gon]]s x 4
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|{{#ifeq:{{{radius|}}}|1|[[{{{wiki|}}}24-cell#Hexagons|Long radius]]|{{#ifeq:{{{radius|}}}|{{radic|2}}|[[{{{wiki|}}}24-cell#Squares|Long radius]]|Long radius}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Edge length{{Efn|A procedure to construct each of these 4-polytopes from the 4-polytope to its left (its predecessor) preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The successor edge length will always be less unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the ''same'' as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.|name=edge length of successor|group=}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{5}{2}} \approx 1.581&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{2}}{\phi} \approx 0.874&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{5}{2}} \approx 1.581&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;2-\phi \approx 0.382&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Short radius
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{1}{4}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{2}}{4} \approx 0.354&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{2}}{2} \approx 0.707&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{2}}{2} \approx 0.707&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{4}} \approx 1.309&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{4}} \approx 1.309&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{4}} \approx 1.309&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Area
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;10\left(\tfrac{5\sqrt{3}}{8}\right) \approx 10.825&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;10\left(\tfrac{5\sqrt{3}}{4}\right) \approx 21.651&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;32\left(\sqrt{\tfrac{3}{4}}\right) \approx 27.713&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;32\left(\sqrt{3}\right) \approx 55.425&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;24&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;48&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;96\left(\sqrt{\tfrac{3}{16}}\right) \approx 41.569&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;96\left(\sqrt{\tfrac{3}{4}}\right) \approx 83.138&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1200\left(\tfrac{\sqrt{3}}{4\phi^2}\right) \approx 198.48&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;1200\left(\tfrac{2\sqrt{3}}{4\phi^2}\right) \approx 396.95&lt;/math&gt;&lt;/small&gt;}}}}
|
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{8\phi^4}\right) \approx 90.366&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{4\phi^4}\right) \approx 180.73&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Volume
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;5\left(\tfrac{5\sqrt{5}}{24}\right) \approx 2.329&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;5\left(\tfrac{5\sqrt{10}}{12}\right) \approx 6.588&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;16\left(\tfrac{1}{3}\right) \approx 5.333&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;16\left(\tfrac{2\sqrt{2}}{3}\right) \approx 15.085&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;8&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;8\sqrt{8} \approx 22.627&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;24\left(\tfrac{\sqrt{2}}{3}\right) \approx 11.314&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;24\left(\tfrac{4}{3}\right) = 32&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;600\left(\tfrac{\sqrt{2}}{12\phi^3}\right) \approx 16.693&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;600\left(\tfrac{4}{12\phi^3}\right) \approx 47.214&lt;/math&gt;&lt;/small&gt;}}}}
|
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6\sqrt{8}}\right) \approx 18.118&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6}\right) \approx 51.246&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|4-Content
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5}}{24}\left(\tfrac{\sqrt{5}}{2}\right)^4 \approx 0.146&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5}}{24}\left(\sqrt{5}\right)^4 \approx 2.329&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{2}{3} \approx 0.667&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{8}{3} \approx 2.666&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;8&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 3.863&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 15.451&lt;/math&gt;&lt;/small&gt;}}}}
|
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 4.193&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 16.770&lt;/math&gt;&lt;/small&gt;}}}}
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{{Information
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== Licensing ==
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{{Information
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|Source=Luz Maria Hernandez-Saenz
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== Licensing ==

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{{Information
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|Source=[[WikiJournal_Preprints/The_effect_of_local_millet_drink_(Kunu)_on_the_testis_and_epididymis_of_adult_male_wistar_rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]] 
|Date=23:53, 23 February 2022 (UTC)
|Author=Darlington Cyprain Akukwu; Godwin Chinedu Uloneme; Damian Nnabuihe Ezejindu; Princewill Sopuluchukwu Udodi; Ifesinachi Ogochukwu Ezejindu; Chukwudi Jesse Nwajagu; Benedict Nzube Obinwa; Ifechukwu Justicia Obiesie; Emeka Christian Okafor; Somadina Nnamdi Okeke; Doris Kasarachi Ogbuokiri; Ambrose Echefulachi Agulanna
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{{Information
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== Licensing ==
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      <text bytes="9783" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240210.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20240330.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240502.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>dj7kdfit2lqmo4eymf7tf8gcv3t4hgu</sha1>
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  </page>
  <page>
    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
    <id>285384</id>
    <revision>
      <id>2624713</id>
      <parentid>2624518</parentid>
      <timestamp>2024-05-02T16:32:51Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Integer Arithmetic */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3044" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Borrow ([[Media:Borrow.20230701.pdf |pdf]])
* Overflow ([[Media:gcc.Overflow.20240502.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Overview ====

* Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]])

* Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]])

* Shared Library Background ([[Media:Link.3.SharedLibrary.20220924.pdf |pdf]])

==== Library Search Path ====

* Library Search Path ([[Media:Link.4.LibrarySearch.20231002.pdf |pdf]])

* Library Search Using -rpath ([[Media:Link.5.LibraryRPATH.20240430.pdf |pdf]])

* Library Search Examples ([[Media:Link.6.LibraryExample.20240501.pdf |pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
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      <contributor>
        <username>Alexis Jazz</username>
        <id>791434</id>
      </contributor>
      <comment>Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]])</comment>
      <model>wikitext</model>
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      <text bytes="1278" xml:space="preserve">{{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}}
The following data is cached, and was last updated 2024-05-01T07:26:12Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache.
{| class=&quot;sortable wikitable&quot;
! Gadget !! data-sort-type=&quot;number&quot; | Number of users !! data-sort-type=&quot;number&quot; | Active users
|-
|CleanDeletions || 75 || 0
|-
|EnhancedTalk || 1352 || 7
|-
|HideFundraisingNotice || 796 || 12
|-
|HotCat || 866 || 13
|-
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|Round Corners || 1145 || 4
|-
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* [[Special:GadgetUsage]]
* [[m:Meta:GUS2Wiki/Script|GUS2Wiki]]
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== Licensing ==
{{Cc-by-4.0}}

==References==
&lt;references group=&quot;lower-alpha&quot;/&gt;
[[Category:WikiJournal]]</text>
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
[[Category:WikiJournal]]</text>
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      <text bytes="372" xml:space="preserve">== Summary ==
{{Information
|Description=PDF version of [[WikiJournal of Science/A Survey on Internet Protocol version 4 (IPv4)|A Survey on Internet Protocol version 4 (IPv4)]]
|source=[[WikiJournal of Science/A Survey on Internet Protocol version 4 (IPv4)]]
|Date=25-12-2022
|Author=Michel Bakni and Sandra Hanbo
}}

== Licensing ==
{{cc-by-4.0}}
[[Category:WikiJournal]]</text>
      <sha1>200ausw1e98dbs084e540xvi7dt4bka</sha1>
    </revision>
  </page>
  <page>
    <title>User:Guy vandegrift/sandbox/Archives/1</title>
    <ns>2</ns>
    <id>291670</id>
    <revision>
      <id>2624860</id>
      <parentid>2556927</parentid>
      <timestamp>2024-05-02T23:38:32Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17642" xml:space="preserve">{{/header/}}
__TOC__
04:11, 1 January 2023 (UTC)
----

==A==
[https://www.google.com/search?q=Socialism+site%3Adocs.google.com&amp;rlz=1C1CHBF_enUS1007US1007&amp;sxsrf=ALiCzsbf_f6YqprH9EPmJ9uYKX0U5vJH_Q%3A1672527697188&amp;ei=Ub-wY_CMC_qeqtsP6bK2sA0&amp;ved=0ahUKEwiw-ZGa-6T8AhV6j2oFHWmZDdYQ4dUDCBE&amp;uact=5&amp;oq=Socialism+site%3Adocs.google.com&amp;gs_lcp=Cgxnd3Mtd2l6LXNlcnAQA0oECEEYAEoECEYYAFAAWMoUYO0baABwAHgAgAFPiAGeBJIBATmYAQCgAQHAAQE&amp;sclient=gws-wiz-serp#ip=1 Socialism Google search with keywords site:docs.google.com]

[https://docs.google.com/document/preview?hgd=1&amp;id=1fzTVcIzElIMqioRVgudahXcSD32v79bz4AKHfVpOQkg Socialism essay stored by Google]

{{dotorg}}

==Beats with lilypond==
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;2 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 4/4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4


} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
                            
&lt;d&gt;4 &lt;aes'&gt;4 &lt;d,&gt;4 &lt;aes'&gt;4|                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2. &lt;aes&gt;4  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===6/8===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;Rest 4 bars; beat 1 bar; rest 8 bars, then 1 beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 6/8 
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
|&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.
} 
\new Staff \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;glockenspiel&quot; \clef treble 
r2. | r2. |
r2. | r2. |
&lt;d&gt;4 &lt;aes'&gt;4 &lt;aes&gt;4 |
r2. | r2. |r2. |r2. |
r2. | r2. |r2. |r2. |
&lt;d,&gt;4 &lt;aes'&gt;4 &lt;aes&gt;4  
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

=== 4/4 ===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;4 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 4/4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ 

} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
r1 | r1 |                                        %2                                   
&lt;d&gt;4 &lt;aes'&gt;4 &lt;d,&gt;4 &lt;aes'&gt;4 |                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2.&lt;d,&gt;4  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===2/2===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;2 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 2/2
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ 

} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
r1 | r1 |                                        %2                                   
&lt;d&gt;2 &lt;aes'&gt;2|                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2 &lt;d,&gt;2  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===unfinished beethoven 7 passage===
&lt;score sound=&quot;1&quot;&gt;\language &quot;english&quot;\new Staff &lt;&lt; \new Voice = &quot;first&quot;\relative {\clef treble\time 3/4 \key d \major \set Score.tempoHideNote = ##t 
\tempo 4 = 228 \voiceOne 
&lt;a' a,&gt;2.~              |a2.          |&lt;a a,&gt;2.~              |~a2.          |&lt;a, d a'&gt;4~&lt;a a' d,&gt;4~&lt;a e' a&gt;|a'4~a4~a4~

}\new Voice= &quot;second&quot;\relative { \voiceTwo 
\override NoteHead.color = #red \override Beam.color = #red \override Accidental.color = #red \override Stem.color = #red \override Rest.color = #red 
d'2 ~d8 cs8           |d4  r4 r4      |d2 ~d8 cs8           |d4  r4 r4       | r4 r4 r4                     | r4 r4 r4              |

}\new Staff \relative{\clef bass\time 3/4  \key d \major
&lt;fs d&gt;2~&lt;fs d&gt;8&lt;e a,&gt;8|&lt;d fs a&gt;4 r4 r4|&lt;fs d&gt;2~&lt;fs d&gt;8&lt;e a,&gt;8|&lt;d fs a&gt;4 r4 r4|&lt;d fs&gt;2 &lt;cs g'&gt;4              |&lt;d fs&gt;4~&lt;cs a'&gt;~&lt;d a'&gt;~
|&lt;a a'&gt;r4 r4
}&gt;&gt;&lt;/score&gt;

See also https://lilypond.org/doc/v2.23/Documentation/notation/simultaneous-notes
==Wikidebates==
Wikidebate collection
===B===
{{special:PrefixIndex/Can electric cars significantly help humanity get off fossil fuels?}}

===D===
{{special:PrefixIndex/Did the United States need to use atomic weapons to end World War II?}}
{{special:PrefixIndex/Do humans have free will?}}
{{special:PrefixIndex/Do living things on Earth have a purpose?}}
{{special:PrefixIndex/Do natural resources exist?}}
{{special:PrefixIndex/Does an individual have a responsibility to maintain a community?}}
{{special:PrefixIndex/Does everything happen for a sufficient reason?}}
{{special:PrefixIndex/Does God exist?}}
{{special:PrefixIndex/Does nuclear power lead to nuclear weapons?}}
{{special:PrefixIndex/Does objective reality exist?}}
{{special:PrefixIndex/Does religion do more harm than good?}}
{{special:PrefixIndex/Does the Catholic Church do more harm than good?}}

===I===
{{special:PrefixIndex/Is a world government desirable?}}
{{special:PrefixIndex/Is aggressive war of territorial expansion good?}}
{{special:PrefixIndex/Is capitalism sustainable?}}
{{special:PrefixIndex/Is collapse of the global civilization before year 2100 likely?}}
{{special:PrefixIndex/Is colonization of Mars in this century realistic?}}
{{special:PrefixIndex/Is morality objective?}}
{{special:PrefixIndex/Is Paleo diet a good thing?}}
{{special:PrefixIndex/Is philosophy any good?}}
{{special:PrefixIndex/Is slavery good?}}
{{special:PrefixIndex/Is the 2022 Russian military operation in Ukraine justified?}}
{{special:PrefixIndex/Is there intelligent extraterrestrial life in the Milky Way?}}
{{special:PrefixIndex/Is Wikipedia consensus process good?}}
===S===
{{Special:PrefixIndex/Should abortion be legal?}}
{{Special:PrefixIndex/Should animal testing be legal?}}
{{Special:PrefixIndex/Should cannabis be legal?}}
{{Special:PrefixIndex/Should capital punishment be legal?}}
{{Special:PrefixIndex/Should civilians be prohibited from owning firearms?}}
{{Special:PrefixIndex/Should cryptocurrencies be banned?}}
{{Special:PrefixIndex/Should infanticide be legal?}}
{{Special:PrefixIndex/Should involuntary treatment be made illegal?}}
{{Special:PrefixIndex/Should it be legal for social media to censor harmful misinformation?}}
{{Special:PrefixIndex/Should Mein Kampf be banned?}}
{{Special:PrefixIndex/Should mentally ill people be allowed to have children?}}
{{Special:PrefixIndex/Should Mill's harm principle be accepted?}}
{{Special:PrefixIndex/Should polygamy be legal?}}
{{Special:PrefixIndex/Should same-sex marriage be legal?}}
{{Special:PrefixIndex/Should sex change operations be guided by mental health specialists or psychologists?}}
{{Special:PrefixIndex/Should suicide be legal?}}
{{Special:PrefixIndex/Should the monarchy in the UK be abolished?}}
{{Special:PrefixIndex/Should the United States have developed the nuclear weapon?}}
{{Special:PrefixIndex/Should the world adopt a one-child policy?}}
{{Special:PrefixIndex/Should Ukraine surrender to Russia in 2022?}}
{{Special:PrefixIndex/Should universal basic income be established?}}
{{Special:PrefixIndex/Should voluntary euthanasia be legal?}}
{{Special:PrefixIndex/Should we aim to reduce the Earth population?}}
{{Special:PrefixIndex/Should we colonize Mars?}}
{{Special:PrefixIndex/Should we go vegan?}}
{{Special:PrefixIndex/Should we have a Wikiversity specific discord server?}}
{{Special:PrefixIndex/Should we merge all WikiJournals into one?}}
{{Special:PrefixIndex/Should we not watch pornography?}}
{{Special:PrefixIndex/Should we use nuclear energy?}}
{{Special:PrefixIndex/Should we use the debate algorithm on wikidebates?}}

===W===
{{Special:PrefixIndex/Was 9/11 an inside job?}}
{{Special:PrefixIndex/Which is the best religion to follow?}}
{{Special:PrefixIndex/Who is Satoshi Nakamoto?}}

===other===
{{Special:PrefixIndex/Wikidebate/Preload}}
{{Special:PrefixIndex/Wikidebate}}

==ogg and mid (midi) files intervals==
12 tone chromatic scale
===Audio template===
{{audio|2ª_d.ogg|P1}}

{{audio|2ª_m.ogg|m2}}

{{audio|2ª M.ogg|M2}}

{{audio|2ª A.ogg|m3}}

{{audio|2 tonos.ogg|m3}}

{{audio|2 tonos, 1 semitono.ogg|P4}}

{{audio|3 tonos.ogg|TT}}

{{audio|3 tonos, 1 semitono.ogg|P5}}

{{audio|4 tonos.ogg|m6}}

{{audio|4 tonos, 1 semitono.ogg|M6}}

{{audio|5 tonos.ogg|m7}}

{{audio|5 tonos, 1 semitono.ogg|M7}}

{{audio|6 tonos, 1 semitono.ogg|P8}}

===Simple links===
{| border=&quot;1&quot;
|+ 12 intervals: Listening to the [[w:Ogg|ogg]] version is less likely to require a file download
! Interval !! Written as !! AKA !! midi !! ogg
|-
!Perfect Unison
|P1
|Unison
|{{audio|Unison.mid}}
|[[file:2ª_d.ogg]]
|-
! Minor Second
| m2
|min 2nd
|{{audio|minor_2.mid}}
|[[file:2ª_m.ogg]]
|-
! Major Second
|M2
|maj 2nd
|{{audio|major_2.mid}}
|[[file:2ª M.ogg]]
|-
! Minor Third
|m3
|min 3rd
|{{audio|minor_3.mid}}
|[[file:2ª A.ogg]]
|-
! Major Third
|M3
|maj 3rd
|{{audio|major_3.mid}}
|[[file:2 tonos.ogg]]
|-
! Perfect Fourth
|P4
|4th
|{{audio|perfect_4.mid}}
|[[file:2 tonos, 1 semitono.ogg]]
|-
!Tritone
|TT
|aug 4, dim 5
|{{audio|tritone.mid}}
|[[file:3 tonos.ogg]]
|-
! Perfect Fifth
|P5
|5th
|{{audio|perfect_5.mid}}
|[[file:3 tonos, 1 semitono.ogg]]
|-
! Minor Sixth
|m6
|min 6th
|{{audio|major_6.mid}} 
|[[file:4 tonos.ogg]]
|-
! Major Sixth
|M6
|maj 6th
|{{audio|minor_6.mid}} 
|[[file:4 tonos, 1 semitono.ogg]]
|-
! Minor Seventh
|m7
|min 7
|{{audio|minor_7.mid}}
|[[file:5 tonos.ogg]]
|-
! Major 7th
|M7
|maj 7
|{{audio|major_7.mid}}
|[[file:5 tonos, 1 semitono.ogg]]
|-
! Perfect Octave
|P8
|8va
|{{audio|octave.mid}}
|[[file:6 tonos.ogg]]
|}

==Listen template==
{{Listen|2ª_d.ogg|P1}}

{{Listen|2ª_m.ogg|m2}}

{{Listen|2ª M.ogg|M2}}

{{Listen|2ª A.ogg|m3}}

{{Listen|2 tonos.ogg|m3}}

{{Listen|2 tonos, 1 semitono.ogg|P4}}

{{Listen|3 tonos.ogg|TT}}

{{Listen|3 tonos, 1 semitono.ogg|P5}}

{{Listen|4 tonos.ogg|m6}}

{{Listen|4 tonos, 1 semitono.ogg|M6}}

{{Listen|5 tonos.ogg|m7}}

{{Listen|5 tonos, 1 semitono.ogg|M7}}

{{Listen|6 tonos, 1 semitono.ogg|P8}}

==experiment==
{{audio|2 tonos.ogg|m3}}

{{Audlisten||2 tonos.ogg|m3}}

[[file:2 tonos.ogg|click]]

==Lilipond Shark attack Beethoven==

[https://www.free-scores.com/download-sheet-music.php?pdf=1212 Beethoven 7th Symphony 3rd Movement orchestral score]
*Pages: 6 (Assai meno presto) and 8 (shark)

[https://youtu.be/JMrm9jEo_Pk?t=1244 Beethoven 7th Symphony Second Movement: Orchestra (Youtube)] 

&lt;big&gt;[https://youtu.be/NaZ49Gl2bHs?t=1626 Piano/violin (Youtube w/score)] [https://youtu.be/NaZ49Gl2bHs?t=1621 Assai meno presto @1621s.] [https://www.youtube.com/watch?v=NaZ49Gl2bHs&amp;t=1626s '''Shark @1626s''')]&lt;/big&gt;

[[https://www.youtube.com/watch?v=EMRwv7qMReo I see a shark]] [https://www.youtube.com/watch?v=M-mAFarwU18 John Williams Shark Theme]


The following is at https://youtu.be/NaZ49Gl2bHs?t=1681
===main===
&lt;score sound=&quot;1&quot;&gt;\language &quot;english&quot;\new Staff &lt;&lt;\new Voice = &quot;first&quot; \clef treble\time 3/4 \key d \major \set Score.tempoHideNote = ##t \relative { \set Staff.midiInstrument = #&quot;electric grand&quot;
\override Score.BarNumber.break-visibility = ##(#t #t #t)\clef treble \tempo 4 = 160 \voiceOne 
|a''2.(  | \grace { b32[( a gs] } a4 cs b)  | a2 (g4~ | g4 b a)  |%5next
g4 (fs a  | d a fs)  |  e2.~  | e4 &lt;a, a'&gt;~ &lt;a a'&gt;8  &lt;gs gs'&gt;8 | &lt;a a'&gt;2.( |  &lt;a a'&gt;2  &lt;a a'&gt;8) &lt;gs gs'&gt;8  |%11next
&lt;a a'&gt;2.~|&lt;a a'&gt;2. | r4 &lt;a b&gt;2 |&lt;a b&gt;2 &lt;a b&gt;4 |r4 &lt;cs bf&gt;2 |&lt;cs bf&gt; &lt;cs bf&gt;4|%17|%18|%19    

}\new Voice= &quot;second&quot;\relative{\voiceTwo 
&lt;a' fs'&gt;2.~(|&lt;a fs'&gt;4 &lt;cs a'&gt; &lt;b g'&gt;) |&lt;a fs'&gt;2 (&lt;g e'&gt;4~ | &lt;g e'&gt; &lt;b g'&gt; &lt;a fs'&gt;)  |%5next
&lt;g e'&gt;4(&lt;fs d'&gt;&lt;a fs'&gt; | &lt;d fs&gt;&lt;a fs'&gt;&lt;fs d'&gt;) | &lt;e cs'&gt;2.~ | &lt;e cs'&gt;4 fs' &lt;fs d&gt; | &lt;cs e&gt;2.( | &lt;cs e&gt;4 fs4 &lt;d fs&gt;4) |%11next
&lt;cs e&gt;4 r r | r2. | r4&lt;ds, fs&gt;2 |&lt;ds fs&gt;2 &lt;ds fs&gt;4  |r4 &lt;g e&gt;2 |&lt;g e&gt; &lt;g e&gt;4|%17|%18|%19      

}{\new Staff &lt;&lt;\new Voice = &quot;third&quot; \relative{\clef bass\time 3/4  \key d \major\clef bass\voiceThree 
&lt;a fs&gt;2.~(|&lt;a fs&gt;4 &lt;cs a&gt; &lt;b g&gt;) | &lt;fs a&gt;2 (&lt;e g&gt;4 | &lt;e g&gt;~ &lt;b' g&gt; &lt;a fs&gt;) |%5next
&lt;e g&gt;4(&lt;d fs&gt;&lt;fs a&gt; | r &lt;fs a&gt;&lt;d fs&gt; | &lt;cs e&gt;2.~ | &lt;cs e&gt;4 &lt;fs a&gt; (&lt;d fs&gt;) | &lt;cs e&gt;2. | &lt;e cs&gt;4 &lt;fs a&gt; &lt;d fs&gt;       |%11next
&lt;cs e&gt;4 r2 | r2. | r2. | r2. |r2. |r2.|%17|%18|%19

}\new Voice= &quot;fourth&quot; \relative{\set Staff.midiInstrument = #&quot;electric grand&quot;\override NoteHead.color = #red \override Beam.color = #red \override Accidental.color = #red \override Stem.color = #red \override Rest.color = #red \override Flag.color = #red \override Slur.color = #red \override Tie.color = #red \voiceFour 
a,2~ (a8 ~gs8 | a4) r4 r4 | a2~ (a8 ~gs8 | a4) r4 r4         |%5next
 a2~ (a8 ~gs8 | a4) r4 r4 |a2~ (a8 ~gs8 | a4) r4 r4 | a2~ (a8 ~gs8 | a4) r4 r4 |%11next
a4 gs (a8) r | gs4(a8)r8 gs4( |a8) r8 gs4(a8) r8 |gs4(a8)r8 gs4( |a8) r8 gs4 (a8) r|gs4(a8)r8 gs4|%17|%18|%19

}&gt;&gt;}&gt;&gt;&lt;/score&gt;

==Polyphony easy-to-read==
&lt;score sound=&quot;1&quot;&gt;{ \new PianoStaff &lt;&lt; %Global-double-bracket START
\relative c'' { \clef treble \tempo 4 = 60 r8 g16 c e g, c e r8 g,16 c e g, c e} 
\new Staff &lt;&lt; %Local-double-bracket START
\relative c' {\clef bass r16 e8. ~e4 r16 e8. ~e4} 
   \new Voice= &quot;first&quot; \relative{\clef bass\voiceOne c'2 c2 } &gt;&gt; %Local-double-bracket END
                                        &gt;&gt; %Global-double-bracket END
}&lt;/score&gt;

-----

&lt;score sound=&quot;1&quot;&gt;
\language &quot;english&quot;\new PianoStaff &lt;&lt; 
  &lt;&lt; \new Voice=&quot;first&quot;\clef treble\relative 
     { \voiceOne \clef treble\tempo 4=160 a4 b c d e d g2 } &gt;&gt;
     { \new Staff  &lt;&lt; \new Voice=&quot;second&quot;\relative{\clef bass\voiceTwo a4 b c d e d g2}
                       \new Voice=&quot;third&quot; \relative{\voiceThree a,4 b c d e d g2}       &gt;&gt; }
       &gt;&gt;&lt;/score&gt;</text>
      <sha1>a1k1e48uimyc74tvau856luwxsbo7hp</sha1>
    </revision>
    <revision>
      <id>2624861</id>
      <parentid>2624860</parentid>
      <timestamp>2024-05-02T23:40:30Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17635" xml:space="preserve">{{/header/}}

04:11, 1 January 2023 (UTC)
----

==A==
[https://www.google.com/search?q=Socialism+site%3Adocs.google.com&amp;rlz=1C1CHBF_enUS1007US1007&amp;sxsrf=ALiCzsbf_f6YqprH9EPmJ9uYKX0U5vJH_Q%3A1672527697188&amp;ei=Ub-wY_CMC_qeqtsP6bK2sA0&amp;ved=0ahUKEwiw-ZGa-6T8AhV6j2oFHWmZDdYQ4dUDCBE&amp;uact=5&amp;oq=Socialism+site%3Adocs.google.com&amp;gs_lcp=Cgxnd3Mtd2l6LXNlcnAQA0oECEEYAEoECEYYAFAAWMoUYO0baABwAHgAgAFPiAGeBJIBATmYAQCgAQHAAQE&amp;sclient=gws-wiz-serp#ip=1 Socialism Google search with keywords site:docs.google.com]

[https://docs.google.com/document/preview?hgd=1&amp;id=1fzTVcIzElIMqioRVgudahXcSD32v79bz4AKHfVpOQkg Socialism essay stored by Google]

{{dotorg}}

==Beats with lilypond==
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;2 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 4/4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4


} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
                            
&lt;d&gt;4 &lt;aes'&gt;4 &lt;d,&gt;4 &lt;aes'&gt;4|                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2. &lt;aes&gt;4  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===6/8===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;Rest 4 bars; beat 1 bar; rest 8 bars, then 1 beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 6/8 
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.~
|&lt;bes f'&gt;2.~ |&lt;bes f'&gt;2.
} 
\new Staff \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;glockenspiel&quot; \clef treble 
r2. | r2. |
r2. | r2. |
&lt;d&gt;4 &lt;aes'&gt;4 &lt;aes&gt;4 |
r2. | r2. |r2. |r2. |
r2. | r2. |r2. |r2. |
&lt;d,&gt;4 &lt;aes'&gt;4 &lt;aes&gt;4  
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

=== 4/4 ===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;4 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 4/4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ 

} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
r1 | r1 |                                        %2                                   
&lt;d&gt;4 &lt;aes'&gt;4 &lt;d,&gt;4 &lt;aes'&gt;4 |                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2.&lt;d,&gt;4  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===2/2===
&lt;score sound=&quot;1&quot;&gt;
\header {piece = &quot;2 beats/bar: Rest 2 bars; beat 1 bar; rest 6 bars; 1-2-3-beat (89.375 beats per minute)&quot;}
{ 
\new GrandStaff &lt;&lt; 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c'' { \set Staff.midiInstrument = #&quot;reed organ&quot; \clef treble \tempo 32 = 715 \time 2/2
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ | %4
&lt;bes f'&gt;1~ |&lt;bes f'&gt;1~ 

} 
\new Staff % \with{ \magnifyStaff #6/7 }
\relative c''' { \set Staff.midiInstrument = #&quot;tinkle bell&quot; \clef treble 
r1 | r1 |                                        %2                                   
&lt;d&gt;2 &lt;aes'&gt;2|                    %1                   
r1 | r1 |r1 | r1 | r1 | r1                 %6 
r2 &lt;d,&gt;2  |                    %1                       
                      } &gt;&gt; }&lt;/score&gt;

The quarter notes in this score represent the rate at which &quot;phase beats&quot; occur.  This was accomplished by establishing the tempo to be &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} (32nd notes per minute.)  This unconventional tempo is a consequence of the fact that LilyPond can only define tempo in terms of integers.  For the pitches used in this score, the equal tempered perfect fifth will produce phase beats at a rate of &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.386}} (beats per minute.)  Since this value is not an integer, we count smaller intervals of time: There are eight 32th notes in a quarter note (because &lt;math&gt;8\times\tfrac{1}{32}=\tfrac 1 4&lt;/math&gt;.)  Since &lt;math&gt;715/8=89.375&lt;/math&gt;, the tempo &lt;big&gt;𝅘𝅥𝅰&lt;/big&gt; ={{math|715}} is exactly the same as &lt;big&gt;&amp;#9833;&lt;/big&gt;={{math|89.375}}, which is only 0.01% slower than the desired tempo.

===unfinished beethoven 7 passage===
&lt;score sound=&quot;1&quot;&gt;\language &quot;english&quot;\new Staff &lt;&lt; \new Voice = &quot;first&quot;\relative {\clef treble\time 3/4 \key d \major \set Score.tempoHideNote = ##t 
\tempo 4 = 228 \voiceOne 
&lt;a' a,&gt;2.~              |a2.          |&lt;a a,&gt;2.~              |~a2.          |&lt;a, d a'&gt;4~&lt;a a' d,&gt;4~&lt;a e' a&gt;|a'4~a4~a4~

}\new Voice= &quot;second&quot;\relative { \voiceTwo 
\override NoteHead.color = #red \override Beam.color = #red \override Accidental.color = #red \override Stem.color = #red \override Rest.color = #red 
d'2 ~d8 cs8           |d4  r4 r4      |d2 ~d8 cs8           |d4  r4 r4       | r4 r4 r4                     | r4 r4 r4              |

}\new Staff \relative{\clef bass\time 3/4  \key d \major
&lt;fs d&gt;2~&lt;fs d&gt;8&lt;e a,&gt;8|&lt;d fs a&gt;4 r4 r4|&lt;fs d&gt;2~&lt;fs d&gt;8&lt;e a,&gt;8|&lt;d fs a&gt;4 r4 r4|&lt;d fs&gt;2 &lt;cs g'&gt;4              |&lt;d fs&gt;4~&lt;cs a'&gt;~&lt;d a'&gt;~
|&lt;a a'&gt;r4 r4
}&gt;&gt;&lt;/score&gt;

See also https://lilypond.org/doc/v2.23/Documentation/notation/simultaneous-notes
==Wikidebates==
Wikidebate collection
===B===
{{special:PrefixIndex/Can electric cars significantly help humanity get off fossil fuels?}}

===D===
{{special:PrefixIndex/Did the United States need to use atomic weapons to end World War II?}}
{{special:PrefixIndex/Do humans have free will?}}
{{special:PrefixIndex/Do living things on Earth have a purpose?}}
{{special:PrefixIndex/Do natural resources exist?}}
{{special:PrefixIndex/Does an individual have a responsibility to maintain a community?}}
{{special:PrefixIndex/Does everything happen for a sufficient reason?}}
{{special:PrefixIndex/Does God exist?}}
{{special:PrefixIndex/Does nuclear power lead to nuclear weapons?}}
{{special:PrefixIndex/Does objective reality exist?}}
{{special:PrefixIndex/Does religion do more harm than good?}}
{{special:PrefixIndex/Does the Catholic Church do more harm than good?}}

===I===
{{special:PrefixIndex/Is a world government desirable?}}
{{special:PrefixIndex/Is aggressive war of territorial expansion good?}}
{{special:PrefixIndex/Is capitalism sustainable?}}
{{special:PrefixIndex/Is collapse of the global civilization before year 2100 likely?}}
{{special:PrefixIndex/Is colonization of Mars in this century realistic?}}
{{special:PrefixIndex/Is morality objective?}}
{{special:PrefixIndex/Is Paleo diet a good thing?}}
{{special:PrefixIndex/Is philosophy any good?}}
{{special:PrefixIndex/Is slavery good?}}
{{special:PrefixIndex/Is the 2022 Russian military operation in Ukraine justified?}}
{{special:PrefixIndex/Is there intelligent extraterrestrial life in the Milky Way?}}
{{special:PrefixIndex/Is Wikipedia consensus process good?}}
===S===
{{Special:PrefixIndex/Should abortion be legal?}}
{{Special:PrefixIndex/Should animal testing be legal?}}
{{Special:PrefixIndex/Should cannabis be legal?}}
{{Special:PrefixIndex/Should capital punishment be legal?}}
{{Special:PrefixIndex/Should civilians be prohibited from owning firearms?}}
{{Special:PrefixIndex/Should cryptocurrencies be banned?}}
{{Special:PrefixIndex/Should infanticide be legal?}}
{{Special:PrefixIndex/Should involuntary treatment be made illegal?}}
{{Special:PrefixIndex/Should it be legal for social media to censor harmful misinformation?}}
{{Special:PrefixIndex/Should Mein Kampf be banned?}}
{{Special:PrefixIndex/Should mentally ill people be allowed to have children?}}
{{Special:PrefixIndex/Should Mill's harm principle be accepted?}}
{{Special:PrefixIndex/Should polygamy be legal?}}
{{Special:PrefixIndex/Should same-sex marriage be legal?}}
{{Special:PrefixIndex/Should sex change operations be guided by mental health specialists or psychologists?}}
{{Special:PrefixIndex/Should suicide be legal?}}
{{Special:PrefixIndex/Should the monarchy in the UK be abolished?}}
{{Special:PrefixIndex/Should the United States have developed the nuclear weapon?}}
{{Special:PrefixIndex/Should the world adopt a one-child policy?}}
{{Special:PrefixIndex/Should Ukraine surrender to Russia in 2022?}}
{{Special:PrefixIndex/Should universal basic income be established?}}
{{Special:PrefixIndex/Should voluntary euthanasia be legal?}}
{{Special:PrefixIndex/Should we aim to reduce the Earth population?}}
{{Special:PrefixIndex/Should we colonize Mars?}}
{{Special:PrefixIndex/Should we go vegan?}}
{{Special:PrefixIndex/Should we have a Wikiversity specific discord server?}}
{{Special:PrefixIndex/Should we merge all WikiJournals into one?}}
{{Special:PrefixIndex/Should we not watch pornography?}}
{{Special:PrefixIndex/Should we use nuclear energy?}}
{{Special:PrefixIndex/Should we use the debate algorithm on wikidebates?}}

===W===
{{Special:PrefixIndex/Was 9/11 an inside job?}}
{{Special:PrefixIndex/Which is the best religion to follow?}}
{{Special:PrefixIndex/Who is Satoshi Nakamoto?}}

===other===
{{Special:PrefixIndex/Wikidebate/Preload}}
{{Special:PrefixIndex/Wikidebate}}

==ogg and mid (midi) files intervals==
12 tone chromatic scale
===Audio template===
{{audio|2ª_d.ogg|P1}}

{{audio|2ª_m.ogg|m2}}

{{audio|2ª M.ogg|M2}}

{{audio|2ª A.ogg|m3}}

{{audio|2 tonos.ogg|m3}}

{{audio|2 tonos, 1 semitono.ogg|P4}}

{{audio|3 tonos.ogg|TT}}

{{audio|3 tonos, 1 semitono.ogg|P5}}

{{audio|4 tonos.ogg|m6}}

{{audio|4 tonos, 1 semitono.ogg|M6}}

{{audio|5 tonos.ogg|m7}}

{{audio|5 tonos, 1 semitono.ogg|M7}}

{{audio|6 tonos, 1 semitono.ogg|P8}}

===Simple links===
{| border=&quot;1&quot;
|+ 12 intervals: Listening to the [[w:Ogg|ogg]] version is less likely to require a file download
! Interval !! Written as !! AKA !! midi !! ogg
|-
!Perfect Unison
|P1
|Unison
|{{audio|Unison.mid}}
|[[file:2ª_d.ogg]]
|-
! Minor Second
| m2
|min 2nd
|{{audio|minor_2.mid}}
|[[file:2ª_m.ogg]]
|-
! Major Second
|M2
|maj 2nd
|{{audio|major_2.mid}}
|[[file:2ª M.ogg]]
|-
! Minor Third
|m3
|min 3rd
|{{audio|minor_3.mid}}
|[[file:2ª A.ogg]]
|-
! Major Third
|M3
|maj 3rd
|{{audio|major_3.mid}}
|[[file:2 tonos.ogg]]
|-
! Perfect Fourth
|P4
|4th
|{{audio|perfect_4.mid}}
|[[file:2 tonos, 1 semitono.ogg]]
|-
!Tritone
|TT
|aug 4, dim 5
|{{audio|tritone.mid}}
|[[file:3 tonos.ogg]]
|-
! Perfect Fifth
|P5
|5th
|{{audio|perfect_5.mid}}
|[[file:3 tonos, 1 semitono.ogg]]
|-
! Minor Sixth
|m6
|min 6th
|{{audio|major_6.mid}} 
|[[file:4 tonos.ogg]]
|-
! Major Sixth
|M6
|maj 6th
|{{audio|minor_6.mid}} 
|[[file:4 tonos, 1 semitono.ogg]]
|-
! Minor Seventh
|m7
|min 7
|{{audio|minor_7.mid}}
|[[file:5 tonos.ogg]]
|-
! Major 7th
|M7
|maj 7
|{{audio|major_7.mid}}
|[[file:5 tonos, 1 semitono.ogg]]
|-
! Perfect Octave
|P8
|8va
|{{audio|octave.mid}}
|[[file:6 tonos.ogg]]
|}

==Listen template==
{{Listen|2ª_d.ogg|P1}}

{{Listen|2ª_m.ogg|m2}}

{{Listen|2ª M.ogg|M2}}

{{Listen|2ª A.ogg|m3}}

{{Listen|2 tonos.ogg|m3}}

{{Listen|2 tonos, 1 semitono.ogg|P4}}

{{Listen|3 tonos.ogg|TT}}

{{Listen|3 tonos, 1 semitono.ogg|P5}}

{{Listen|4 tonos.ogg|m6}}

{{Listen|4 tonos, 1 semitono.ogg|M6}}

{{Listen|5 tonos.ogg|m7}}

{{Listen|5 tonos, 1 semitono.ogg|M7}}

{{Listen|6 tonos, 1 semitono.ogg|P8}}

==experiment==
{{audio|2 tonos.ogg|m3}}

{{Audlisten||2 tonos.ogg|m3}}

[[file:2 tonos.ogg|click]]

==Lilipond Shark attack Beethoven==

[https://www.free-scores.com/download-sheet-music.php?pdf=1212 Beethoven 7th Symphony 3rd Movement orchestral score]
*Pages: 6 (Assai meno presto) and 8 (shark)

[https://youtu.be/JMrm9jEo_Pk?t=1244 Beethoven 7th Symphony Second Movement: Orchestra (Youtube)] 

&lt;big&gt;[https://youtu.be/NaZ49Gl2bHs?t=1626 Piano/violin (Youtube w/score)] [https://youtu.be/NaZ49Gl2bHs?t=1621 Assai meno presto @1621s.] [https://www.youtube.com/watch?v=NaZ49Gl2bHs&amp;t=1626s '''Shark @1626s''')]&lt;/big&gt;

[[https://www.youtube.com/watch?v=EMRwv7qMReo I see a shark]] [https://www.youtube.com/watch?v=M-mAFarwU18 John Williams Shark Theme]


The following is at https://youtu.be/NaZ49Gl2bHs?t=1681
===main===
&lt;score sound=&quot;1&quot;&gt;\language &quot;english&quot;\new Staff &lt;&lt;\new Voice = &quot;first&quot; \clef treble\time 3/4 \key d \major \set Score.tempoHideNote = ##t \relative { \set Staff.midiInstrument = #&quot;electric grand&quot;
\override Score.BarNumber.break-visibility = ##(#t #t #t)\clef treble \tempo 4 = 160 \voiceOne 
|a''2.(  | \grace { b32[( a gs] } a4 cs b)  | a2 (g4~ | g4 b a)  |%5next
g4 (fs a  | d a fs)  |  e2.~  | e4 &lt;a, a'&gt;~ &lt;a a'&gt;8  &lt;gs gs'&gt;8 | &lt;a a'&gt;2.( |  &lt;a a'&gt;2  &lt;a a'&gt;8) &lt;gs gs'&gt;8  |%11next
&lt;a a'&gt;2.~|&lt;a a'&gt;2. | r4 &lt;a b&gt;2 |&lt;a b&gt;2 &lt;a b&gt;4 |r4 &lt;cs bf&gt;2 |&lt;cs bf&gt; &lt;cs bf&gt;4|%17|%18|%19    

}\new Voice= &quot;second&quot;\relative{\voiceTwo 
&lt;a' fs'&gt;2.~(|&lt;a fs'&gt;4 &lt;cs a'&gt; &lt;b g'&gt;) |&lt;a fs'&gt;2 (&lt;g e'&gt;4~ | &lt;g e'&gt; &lt;b g'&gt; &lt;a fs'&gt;)  |%5next
&lt;g e'&gt;4(&lt;fs d'&gt;&lt;a fs'&gt; | &lt;d fs&gt;&lt;a fs'&gt;&lt;fs d'&gt;) | &lt;e cs'&gt;2.~ | &lt;e cs'&gt;4 fs' &lt;fs d&gt; | &lt;cs e&gt;2.( | &lt;cs e&gt;4 fs4 &lt;d fs&gt;4) |%11next
&lt;cs e&gt;4 r r | r2. | r4&lt;ds, fs&gt;2 |&lt;ds fs&gt;2 &lt;ds fs&gt;4  |r4 &lt;g e&gt;2 |&lt;g e&gt; &lt;g e&gt;4|%17|%18|%19      

}{\new Staff &lt;&lt;\new Voice = &quot;third&quot; \relative{\clef bass\time 3/4  \key d \major\clef bass\voiceThree 
&lt;a fs&gt;2.~(|&lt;a fs&gt;4 &lt;cs a&gt; &lt;b g&gt;) | &lt;fs a&gt;2 (&lt;e g&gt;4 | &lt;e g&gt;~ &lt;b' g&gt; &lt;a fs&gt;) |%5next
&lt;e g&gt;4(&lt;d fs&gt;&lt;fs a&gt; | r &lt;fs a&gt;&lt;d fs&gt; | &lt;cs e&gt;2.~ | &lt;cs e&gt;4 &lt;fs a&gt; (&lt;d fs&gt;) | &lt;cs e&gt;2. | &lt;e cs&gt;4 &lt;fs a&gt; &lt;d fs&gt;       |%11next
&lt;cs e&gt;4 r2 | r2. | r2. | r2. |r2. |r2.|%17|%18|%19

}\new Voice= &quot;fourth&quot; \relative{\set Staff.midiInstrument = #&quot;electric grand&quot;\override NoteHead.color = #red \override Beam.color = #red \override Accidental.color = #red \override Stem.color = #red \override Rest.color = #red \override Flag.color = #red \override Slur.color = #red \override Tie.color = #red \voiceFour 
a,2~ (a8 ~gs8 | a4) r4 r4 | a2~ (a8 ~gs8 | a4) r4 r4         |%5next
 a2~ (a8 ~gs8 | a4) r4 r4 |a2~ (a8 ~gs8 | a4) r4 r4 | a2~ (a8 ~gs8 | a4) r4 r4 |%11next
a4 gs (a8) r | gs4(a8)r8 gs4( |a8) r8 gs4(a8) r8 |gs4(a8)r8 gs4( |a8) r8 gs4 (a8) r|gs4(a8)r8 gs4|%17|%18|%19

}&gt;&gt;}&gt;&gt;&lt;/score&gt;

==Polyphony easy-to-read==
&lt;score sound=&quot;1&quot;&gt;{ \new PianoStaff &lt;&lt; %Global-double-bracket START
\relative c'' { \clef treble \tempo 4 = 60 r8 g16 c e g, c e r8 g,16 c e g, c e} 
\new Staff &lt;&lt; %Local-double-bracket START
\relative c' {\clef bass r16 e8. ~e4 r16 e8. ~e4} 
   \new Voice= &quot;first&quot; \relative{\clef bass\voiceOne c'2 c2 } &gt;&gt; %Local-double-bracket END
                                        &gt;&gt; %Global-double-bracket END
}&lt;/score&gt;

-----

&lt;score sound=&quot;1&quot;&gt;
\language &quot;english&quot;\new PianoStaff &lt;&lt; 
  &lt;&lt; \new Voice=&quot;first&quot;\clef treble\relative 
     { \voiceOne \clef treble\tempo 4=160 a4 b c d e d g2 } &gt;&gt;
     { \new Staff  &lt;&lt; \new Voice=&quot;second&quot;\relative{\clef bass\voiceTwo a4 b c d e d g2}
                       \new Voice=&quot;third&quot; \relative{\voiceThree a,4 b c d e d g2}       &gt;&gt; }
       &gt;&gt;&lt;/score&gt;</text>
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==Other objects in here==
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==Other objects in here==
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{{Information
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|Source=Eleanor Shaw
|Date=10 January 2023
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==Licensing==
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{{Information
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== Licensing ==
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== Licensing ==
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{{Information
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|Source=https://doi.org/10.15347/WJM/2023.001
|Date=20 Jan 2023
|Author=Emily Pender, Liana Kostak, Kelsey Sutton, Cody Naccarato, Angelina Tsai, Tammy Chung, Stacey Daughters
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== Licensing ==
{{cc-by-4.0}}
[[Category:WikiJournal]]</text>
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      <text bytes="855" xml:space="preserve">My name is Holger Brenner and I am a professor of mathematics in Osnabrück (before that I was a lecturer/reader at the University of Sheffield). For my mathematical courses in German see [[v:de:Benutzer:Holger_Brenner/Lehre]]. My research interests is in algebraic geometry and commutative algebra, in particular closure operation for ideals (tight closure, Hilbert-Kunz theory), vector bundles, Grothendieck-topologies, algebraic differential operators and invariant theory. 

Teaching this term

[[Mathematics for Applied Sciences (Osnabrück_2023-2024)/Part I]] 

Lecture series

[[Vector bundles and tight closure (Triest 2023)]]

Related series held in the past.

[[Vector bundles and ideal closure operations (MSRI 2012)]]

[[Computation of tight closure (Ann Arbor 2012)]]

[[Vector bundles, forcing algebras and local cohomology (Medellin 2012)]]</text>
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== Licensing ==
{{Cc-by-4.0}}

==References==
&lt;references group=&quot;lower-alpha&quot;/&gt;
[[Category:WikiJournal]]</text>
      <sha1>sk4b9m6ue5pl9jw0i1av6xlvkrmudeu</sha1>
    </revision>
  </page>
  <page>
    <title>User:MGA73/Sandbox</title>
    <ns>2</ns>
    <id>294691</id>
    <revision>
      <id>2624616</id>
      <parentid>2624427</parentid>
      <timestamp>2024-05-02T14:25:20Z</timestamp>
      <contributor>
        <username>MGA73</username>
        <id>137394</id>
      </contributor>
      <comment>WikiJournal</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1325" xml:space="preserve">WikiJournal

# [[:File:BeFunky_carney_journals.jpeg.jpeg]]
# [[:File:Crankshaft_journal_bearing_design.pdf]]
# [[:File:T_Circumcincta_WikiJournal_review_Joaquin_M._Prada.pdf]]
# [[:File:The_aims_and_scope_of_WikiJournal_of_Science.pdf]]
# [[:File:WikiJournal_Bioclogging_-_ES.pdf]]
# [[:File:WikiJournal_PreprintJN_-_Jane_Noyes.pdf]]
# [[:File:WikiJournal_Preprints_-_Ankita_-_Statistical_Review.pdf]]
# [[:File:WikiJournal_Preprints_-_Peer_review_2_-_Ankita_responses.pdf]]
# [[:File:WikiJournal_Preprints_COVID-19_ELIMINATION_AND_CELL_DIFFERENTIATION_-_Wikiversity.pdf]]
# [[:File:WikiJournal_Preprints_Hepatitis_E_corr._pischke.pdf]]
# [[:File:WikiJournal_Preprints_Multiple_object_tracking_-_Todd_Horowitz.pdf]]
# [[:File:WikiJournal_Preprints_Orientia_tsutsugamushi_line_numbered.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R2_edits.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R3_edits.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R4_edits.pdf]]
# [[:File:WikiJournal_of_Medicine,_the_first_Wikipedia-integrated_academic_journal.pdf]]
# [[:File:WikiJournal_of_Medicine_articles_and_citations.jpg]]
# [[:File:WikiJournal_of_Science.Review.Jan12.2018_LEAD_env.pdf]]
# [[:File:WikiJournal_review_-_Dan_Bressington.pdf]]
# [[:File:Wikijournal_of_Science_review_-_T_circumcincta_-_Valentina_Busin.pdf]]</text>
      <sha1>5ba8xqrbhwhbg3kgk03ywhsqinj3av4</sha1>
    </revision>
    <revision>
      <id>2624617</id>
      <parentid>2624616</parentid>
      <timestamp>2024-05-02T14:26:33Z</timestamp>
      <contributor>
        <username>MGA73</username>
        <id>137394</id>
      </contributor>
      <comment>rm 2 links not wikijournal</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1315" xml:space="preserve">WikiJournal

# File:BeFunky_carney_journals.jpeg.jpeg
# File:Crankshaft_journal_bearing_design.pdf
# [[:File:T_Circumcincta_WikiJournal_review_Joaquin_M._Prada.pdf]]
# [[:File:The_aims_and_scope_of_WikiJournal_of_Science.pdf]]
# [[:File:WikiJournal_Bioclogging_-_ES.pdf]]
# [[:File:WikiJournal_PreprintJN_-_Jane_Noyes.pdf]]
# [[:File:WikiJournal_Preprints_-_Ankita_-_Statistical_Review.pdf]]
# [[:File:WikiJournal_Preprints_-_Peer_review_2_-_Ankita_responses.pdf]]
# [[:File:WikiJournal_Preprints_COVID-19_ELIMINATION_AND_CELL_DIFFERENTIATION_-_Wikiversity.pdf]]
# [[:File:WikiJournal_Preprints_Hepatitis_E_corr._pischke.pdf]]
# [[:File:WikiJournal_Preprints_Multiple_object_tracking_-_Todd_Horowitz.pdf]]
# [[:File:WikiJournal_Preprints_Orientia_tsutsugamushi_line_numbered.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R2_edits.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R3_edits.pdf]]
# [[:File:WikiJournal_Preprints_Phage_Therapy-R4_edits.pdf]]
# [[:File:WikiJournal_of_Medicine,_the_first_Wikipedia-integrated_academic_journal.pdf]]
# [[:File:WikiJournal_of_Medicine_articles_and_citations.jpg]]
# [[:File:WikiJournal_of_Science.Review.Jan12.2018_LEAD_env.pdf]]
# [[:File:WikiJournal_review_-_Dan_Bressington.pdf]]
# [[:File:Wikijournal_of_Science_review_-_T_circumcincta_-_Valentina_Busin.pdf]]</text>
      <sha1>gjamnkg78u8x9yap71skv664gzfby1p</sha1>
    </revision>
    <revision>
      <id>2624727</id>
      <parentid>2624617</parentid>
      <timestamp>2024-05-02T16:43:29Z</timestamp>
      <contributor>
        <username>MGA73</username>
        <id>137394</id>
      </contributor>
      <comment>WikiJournal related?</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5464" xml:space="preserve">WikiJournal related?

[[:File:070719_Calculation_of_vLD.tif]]
[[:File:070719_VCC_Plate_Configuration.tif]]
[[:File:070719_Virtual_Lethal_Doses.tif]]
[[:File:091719_Row_B_Growth_Curves.tif]]
[[:File:091719_Row_F_Growth_Curves.tif]]
[[:File:091719_Row_G_Calibration.tif]]
[[:File:091719_Virtual_Survival.tif]]
[[:File:1tim.png]]
[[:File:2018-12-09-Chan_Review_for_Borg.pdf]]
[[:File:Androgen_backdoor_pathway_R1_tracked_changes.pdf]]
[[:File:Baryonyx_Wikimedia_Hendrickx_review.pdf]]
[[:File:Beak_and_Feather_Disease_Virus_-_David_Phalen.pdf]]
[[:File:Beak_and_feather_disease_Rev_review-1_06122020.pdf]]
[[:File:Bioclogging_comments.pdf]]
[[:File:COVID-19_ELIMINATION_AND_CELL_DIFFERENTIATION_(1).pdf]]
[[:File:Comments_on_Design_effect_articleAnonymized.pdf]]
[[:File:Dioxins_and_dioxin-like_compounds_review_-_R3.pdf]]
[[:File:Dioxins_and_dioxin-like_compounds_table_-_R3.pdf]]
[[:File:Dr._Imad_Ghamloush_Notes,_Orhan_Gazi_(first_statesman).pdf]]
[[:File:E-Extension_Nepal_Review_2.pdf]]
[[:File:Eco_Inoculum_Effect_022013_Experiment_6.tif]]
[[:File:Eco_Inoculum_Effect_022513_Experiment_4.tif]]
[[:File:Eco_Inoculum_Effect_022813_Experiment_5.tif]]
[[:File:Eco_Inoculum_Effect_031813_Experiment_2.tif]]
[[:File:Eco_Inoculum_Effect_032113_Experiment_3.tif]]
[[:File:Eco_Inoculum_Effect_Composite_022220.tif]]
[[:File:Eco_Inoculum_Effect_Composite_Virtual_Lethal_Doses.tif]]
[[:File:Ed_Baker.jpg]]
[[:File:Endometrial_cancer_reviewer_2.pdf]]
[[:File:Evolved_human_male_preferences_for_female_body_shape_-_tracked_changes.pdf]]
[[:File:Evolved_human_male_preferences_for_female_body_shape_-_tracked_changes_2.pdf]]
[[:File:Fig_1_Androgen_backdoor_pathway_R1_recommended_changes.pdf]]
[[:File:Flow_chart_of_would-have-been-benefited.jpg]]
[[:File:Grhl3_KO_skull_comparison.PNG]]
[[:File:Health_and_GDP_data.png]]
[[:File:Hepatitis_E_-_María_Teresa_Pérez_Gracia_Revision.pdf]]
[[:File:Human_resources_for_health_in_four_countries.png]]
[[:File:Isfahani_-_Adam_Talib.pdf]]
[[:File:Isfahani_-_Author_corrected.pdf]]
[[:File:Isfahani_-_Hilary_Kilpatrick.pdf]]
[[:File:Kunu_and_wistar_rates_R1.pdf]]
[[:File:Kunu_and_wistar_rates_after_review_tracked_changes.pdf]]
[[:File:Locomotor_play_in_leopard_gecko_-_Second_Reviewer_comments.pdf]]
[[:File:Locomotor_play_in_leopard_gecko_-_Second_Reviewer_comments_RD.pdf]]
[[:File:Locomotor_play_in_leopard_gecko_-_Wolf_Huetteroth.pdf]]
[[:File:Locomotor_play_in_leopard_gecko_-_Wolf_Huetteroth_RD_responses.pdf]]
[[:File:Lysenin_submission_copy_edit_JN_15.2.19.pdf]]
[[:File:Melioidosis_-_David_Dance.pdf]]
[[:File:Melioidosis_-_David_Dance_(replied_comments).pdf]]
[[:File:Myxomatosis_Review_-_Justine_Philip.pdf]]
[[:File:Myxomatosis_Review_-_Morgan_Kain.pdf]]
[[:File:Nice_state_history_ES_notes_-_Reviewer_1.pdf]]
[[:File:Normal_cervix.jpg]]
[[:File:Notes_by_Dr._Rama_Draz_Regarding_Orhan_Gazi,_the_first_statesman_(Original_and_Translated).pdf]]
[[:File:PENICILLIN_ART.edited.pdf]]
[[:File:Peer_review_for_Rotavirus.pdf]]
[[:File:Peer_review_of_cervical_screening_article.pdf]]
[[:File:Preprint_-_A_card_game_for_Bell's_theorem_and_its_loopholes_(Reviewer_3).pdf]]
[[:File:Preprint_-_A_card_game_for_Bell's_theorem_and_its_loopholes_(Reviewer_3_further_comments).pdf]]
[[:File:Preprint_-_ShK_toxin_history,_structure_and_therapeutic_applications_for_autoimmune_diseasesHW.pdf]]
[[:File:Preprint_-_ShK_toxin_history,_structure_and_therapeutic_applications_for_autoimmune_diseases_-_Author_response.pdf]]
[[:File:RIGL_Review_report-Teunis_Geijtenbeek.pdf]]
[[:File:Rehumanise.pdf]]
[[:File:Review_Wiki_Phage_-_Fe_Br.pdf]]
[[:File:Review_by_Hassan_Hallak_-_Article_Osman,_father_of_kings.pdf]]
[[:File:Review_by_Hassan_Hallak_-_Article_Osman,_father_of_kings_(original_handwritten).pdf]]
[[:File:Review_by_Rama_Draz_-_Article_Osman,_father_of_kings.pdf]]
[[:File:Soc_license_forestry_NA_Annotated_text_and_reviewers_comments_-_Ian_Thomson.pdf]]
[[:File:Spaces_arrows.svg]]
[[:File:TIM_review_Cristina_Elisa_Martina.pdf]]
[[:File:TIM_review_Robert_Matthews_.pdf]]
[[:File:TIM_topology.png]]
[[:File:T_Circumcincta_WikiJournal_review_Joaquin_M._Prada.pdf]]
[[:File:The_aims_and_scope_of_WikiJournal_of_Science.pdf]]
[[:File:VCC_Technical_Difficulties.tif]]
[[:File:VCC_flowchart.tif]]
[[:File:VCC_review_-_Jen_Payne.pdf]]
[[:File:VitD_CAP_Wiki_J_Med_2017_--_Reviewer_3_comments.pdf]]
[[:File:VitD_CAP_Wiki_J_Med_2017_--_initial_Editor_feedback.pdf]]
[[:File:WikiJournal_Bioclogging_-_ES.pdf]]
[[:File:WikiJournal_PreprintJN_-_Jane_Noyes.pdf]]
[[:File:WikiJournal_Preprints_-_Ankita_-_Statistical_Review.pdf]]
[[:File:WikiJournal_Preprints_-_Peer_review_2_-_Ankita_responses.pdf]]
[[:File:WikiJournal_Preprints_Hepatitis_E_corr._pischke.pdf]]
[[:File:WikiJournal_Preprints_Multiple_object_tracking_-_Todd_Horowitz.pdf]]
[[:File:WikiJournal_Preprints_Orientia_tsutsugamushi_line_numbered.pdf]]
[[:File:WikiJournal_Preprints_Phage_Therapy-R2_edits.pdf]]
[[:File:WikiJournal_Preprints_Phage_Therapy-R3_edits.pdf]]
[[:File:WikiJournal_Preprints_Phage_Therapy-R4_edits.pdf]]
[[:File:WikiJournal_of_Science.Review.Jan12.2018_LEAD_env.pdf]]
[[:File:WikiJournal_review_-_Dan_Bressington.pdf]]
[[:File:Wiki_review_Purssell.pdf]]
[[:File:Wikijournal_of_Science_review_-_T_circumcincta_-_Valentina_Busin.pdf]]
[[:File:Wikiversity_-_Design_Effect_Review_Article_2023_-_DiSogra.pdf]]
[[:File:Wikiversity_-_Design_Effect_Review_Article_Jan_3,_2024_-_DiSogra_-_Charles_DiSogra.pdf]]
[[:File:Æthelflæd,_Lady_of_the_Mercians_-_reviewer_2.pdf]]</text>
      <sha1>98klbnvk32klfnabzwsetnsojp9l69r</sha1>
    </revision>
    <revision>
      <id>2624775</id>
      <parentid>2624727</parentid>
      <timestamp>2024-05-02T18:37:21Z</timestamp>
      <contributor>
        <username>MGA73</username>
        <id>137394</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5514" xml:space="preserve">WikiJournal related?

[[:File:A_Survey_on_Internet_Protocol_version_4_(IPv4).pdf]]
[[:File:A_broad_introduction_to_RNA-Seq.pdf]]
[[:File:A_card_game_for_Bell's_theorem_and_its_loopholes.pdf]]
[[:File:Abū_al-Faraj_ʿAlī_b._al-Ḥusayn_al-Iṣfahānī,_the_Author_of_the_Kitāb_al-Aghānī.pdf]]
[[:File:Acute_gastrointestinal_bleeding_from_a_chronic_cause_a_teaching_case_report.pdf]]
[[:File:Affine_symmetric_group.pdf]]
[[:File:Allogeneic_component_to_overcome_rejection_in_interspecific_pregnancy.pdf]]
[[:File:Alternative_androgen_pathways.pdf]]
[[:File:An_epidemiology-based_and_a_likelihood_ratio-based_method_of_differential_diagnosis.pdf]]
[[:File:An_overview_of_Lassa_fever.pdf]]
[[:File:Anthracyclines.pdf]]
[[:File:Arabinogalactan-proteins.pdf]]
[[:File:Baryonyx.pdf]]
[[:File:Beak_and_feather_disease_virus_biology_and_resultant_disease.pdf]]
[[:File:Binary_search_algorithm.pdf]]
[[:File:Caesarean_section_photography.pdf]]
[[:File:Can_each_number_be_specified_by_a_finite_text?.pdf]]
[[:File:Cell_disassembly_during_apoptosis.pdf]]
[[:File:Comparison_between_the_Lund-Browder_chart_and_the_BurnCase_3D®_for_consistency_in_estimating_total_body_surface_area_burned.pdf]]
[[:File:Definable.pdf]]
[[:File:Diagram_of_the_pathways_of_human_steroidogenesis.pdf]]
[[:File:Dioxins_and_dioxin-like_compounds_toxicity_in_humans_and_animals,_sources,_and_behaviour_in_the_environment.pdf]]
[[:File:Dyslexia.pdf]]
[[:File:E-extension_in_Nepal_brief_overview_in_Nepalese_agriculture.pdf]]
[[:File:ELEPHANTYEAR.NOV2013.pdf]]
[[:File:Earth-grazing_meteoroid_of_13_October_1990.pdf]]
[[:File:Emotional_and_Psychological_Impact_of_Interpreting_for_Clients_with_Traumatic_Histories_on_interpreters_a_review_of_qualitative_articles.pdf]]
[[:File:Epidemiology_of_the_Hepatitis_D_virus.pdf]]
[[:File:Establishment_and_clinical_use_of_reference_ranges.pdf]]
[[:File:Estimating_the_lost_benefits_of_not_implementing_a_visual_inspection_with_acetic_acid_screen_and_treat_strategy_for_cervical_cancer_prevention_in_South_Africa.pdf]]
[[:File:Eukaryotic_and_prokaryotic_gene_structure.pdf]]
[[:File:Extract_of_Laurus_nobilis_attenuates_inflammation_and_epithelial_ulcerations_in_an_experimental_model_of_inflammatory_bowel_disease.pdf]]
[[:File:Grainyhead-like_Genes_in_Regulating_Development_and_Genetic_Defects.pdf]]
[[:File:Hepatitis_E.pdf]]
[[:File:Hilda_Rix_Nicholas.pdf]]
[[:File:Ice_drilling_methods.pdf]]
[[:File:Images_of_Aerococcus_urinae.pdf]]
[[:File:Impact_of_xenogenic_mesenchymal_stem_cells_secretome_on_a_humoral_component_of_the_immune_system.pdf]]
[[:File:Insights_into_abdominal_pregnancy.pdf]]
[[:File:Kunu_and_wistar_rates_after_review_tracked_changes.pdf]]
[[:File:Lead_properties,_history,_and_applications.pdf]]
[[:File:Linearized_relativity_Vandegrift_30_pages.pdf]]
[[:File:Loveday,_1458.pdf]]
[[:File:Lysenin.pdf]]
[[:File:Lysine_biosynthesis,_catabolism_and_roles.pdf]]
[[:File:Mealtime_difficulty_in_older_people_with_dementia.pdf]]
[[:File:Medical_gallery_of_Blausen_Medical_2014.pdf]]
[[:File:Medical_gallery_of_David_Richfield_2014.pdf]]
[[:File:Medical_gallery_of_Mikael_Häggström_2014.pdf]]
[[:File:Orientia_tsutsugamushi,_the_agent_of_scrub_typhus.pdf]]
[[:File:Osman_I,_father_of_kings.pdf]]
[[:File:Paragogy-final.pdf]]
[[:File:Paranthodon.pdf]]
[[:File:Peer_review_of_cervical_screening_article.pdf]]
[[:File:Peripatric_speciation.pdf]]
[[:File:Perspectives_on_the_social_license_of_the_forest_products_industry_from_rural_Michigan,_United_States.pdf]]
[[:File:Phage_Therapy.pdf]]
[[:File:Plasmodium_falciparum_erythrocyte_membrane_protein_1.pdf]]
[[:File:Poster_format_of_2015_cervical_screening_article.pdf]]
[[:File:Preprint_-_A_card_game_for_Bell's_theorem_and_its_loopholes_(Reviewer_3).pdf]]
[[:File:RIG-I_like_receptors.pdf]]
[[:File:Radiocarbon_dating.pdf]]
[[:File:Readability_of_English_Wikipedia's_health_information_over_time.pdf]]
[[:File:Reference_ranges_for_estradiol,_progesterone,_luteinizing_hormone_and_follicle-stimulating_hormone_during_the_menstrual_cycle.pdf]]
[[:File:Resources_for_the_Assessment_and_Treatment_of_Substance_Use_Disorder_in_Adolescents.pdf]]
[[:File:Rosetta_Stone.pdf]]
[[:File:Rotavirus.pdf]]
[[:File:ShK_toxin_history,_structure_and_therapeutic_applications_for_autoimmune_diseases.pdf]]
[[:File:Soc_license_forestry_NA_Annotated_text_and_reviewers_comments_-_Ian_Thomson.pdf]]
[[:File:Spaces_in_mathematics.pdf]]
[[:File:Structural_Model_of_Bacteriophage_T4.pdf]]
[[:File:Table_of_pediatric_medical_conditions_and_findings_named_after_foods.pdf]]
[[:File:Teladorsagia_circumcincta.pdf]]
[[:File:The_Cerebellum.pdf]]
[[:File:The_Hippocampus.pdf]]
[[:File:The_Kivu_Ebola_Epidemic.pdf]]
[[:File:The_TIM_barrel_fold.pdf]]
[[:File:The_Year_of_the_Elephant.pdf]]
[[:File:The_aims_and_scope_of_WikiJournal_of_Science.pdf]]
[[:File:Themes_in_Maya_Angelou's_autobiographies.pdf]]
[[:File:Tubal_pregnancy_with_embryo.pdf]]
[[:File:Ultrasonography_of_a_cervical_pregnancy.pdf]]
[[:File:Viewer_interaction_with_YouTube_videos_about_hysterectomy_recovery.pdf]]
[[:File:Virtual_colony_count.pdf]]
[[:File:Vitamin_D_as_an_adjunct_for_acute_community-acquired_pneumonia_among_infants_and_children_systematic_review_and_meta-analysis.pdf]]
[[:File:Western_African_Ebola_virus_epidemic.pdf]]
[[:File:Widgiemoolthalite.pdf]]
[[:File:WikiJournal_of_Medicine,_the_first_Wikipedia-integrated_academic_journal.pdf]]
[[:File:Working_with_Bipolar_Disorder_During_the_COVID-19_Pandemic_Both_Crisis_and_Opportunity.pdf]]
[[:File:Æthelflæd,_Lady_of_the_Mercians.pdf]]</text>
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      <text bytes="362" xml:space="preserve">== Summary ==
{{Information
|Description= WikiJournal of Medicine/Alternative androgens pathways
|Source= 10.15347/WJM/2023.003
|Date= 3rd May 2023
|Author= Maxim G Masiutin and Maneesh K Yadav
|Permission= This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported Licence
}}

== Licensing ==
{{Cc-by-sa-3.0}}
[[Category:WikiJournal]]</text>
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{{Information
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|Date= 8 Apr 2023
|Author= Dhananjay Bhattacharyya, Abhijit Mitra
|Permission= 
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== Licensing ==
{{cc-by-4.0}}
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{{Information
|Description= English: [[WikiJournal of Science/Multiple object tracking]]
|Source= 10.15347/WJS/2023.003
|Date= 12/5/2023
|Author= Alex O. Holcombe
|Permission= This file is licensed under the creative commons attribution share-alike 3.0 unported license.
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{{Information
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|Source= Bassem Fleifel
|Date= 2023-06-28
|Author= Bassem Fleifel
|Permission= See below
}}

== Licensing ==
{{cc-by-4.0}}
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{{Information
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|Source=[[WikiJournal of Science]]
|Date= 24/5/2023
|Author= Charles DiSogra
|Permission=
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==Licensing==
{{cc-by-4.0}}
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{{Information
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== Licensing ==
{{self|GFDL|cc-by-4.0}}
[[Category:WikiJournal]]</text>
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{{Information
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|Source= Doi: 10.15347/wjh/2023.001
|Date= 20/06/2023
|Author= Ed
|Permission=
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== Licensing ==
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      <text bytes="295" xml:space="preserve">== Summary ==

{{Information
|Description = Peer review note of [[WikiJournal Preprints/Orhan Gazi, the first statesman]] by Dr Imad Ghamloush 
|Source = Bassem Fleifel
|Date = 2023-06-28
|Author = Bassem Fleifel
|Permission = See Below
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== Licensing ==
{{cc-by-4.0}}
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{{Information
|Description= English: Wikijournal of Medicine/Impact of xenogenic mesenchymal stem cells secretome on a humoral component of the immune system
|Source= 10.15347/WJM/2023.004
|Date= 31/07/2023
|Author= Vitalii Moskalov, Olena Koshova, Sabina Ali, Nataliia Filimonova, Irina Tishchenko  
|Permission= This file is licensed under the Creative Commons Attribution-ShareAlike 4.0 International
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== Licensing ==
{{cc-by-4.0}}

[[Category:Immune system]]
[[Category:WikiJournal]]</text>
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{{Information
|Description=Review comments by Edwin Saavedra Cifuentes on [[WikiJournal Preprints/Bioclogging]]
|Source=Submitted by author
|Date=2023-08-09
|Author=Edwin Saavedra C
|Permission=
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== Licensing ==
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  <page>
    <title>Noble Boolean functions</title>
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      <text bytes="8395" xml:space="preserve">{{Zhegalkin stuff}}
Noble Boolean functions are those who are their own [[Zhegalkin twins]], i.e. the binary expression of their [[Algebraic normal form|ANF]] is equal to their truth table.&lt;br&gt;
They correspond to {{w|Fixed point (mathematics)|fixed points}} in the [[Zhegalkin matrix#Zhegalkin permutation|Zhegalkin permutation]].&lt;br&gt;
They are all even, i.e. the first digit of their truth table is false.&lt;br&gt;
When a Boolean function is noble, its whole [[Boolf-EC#P|faction]] is noble.&lt;br&gt;
&lt;small style=&quot;opacity: .5;&quot;&gt;Within the [[Studies of Boolean functions|this project]] it is slightly misleading to apply the term ''noble'' to [[Boolf-term#BF|Boolean functions]]. It is a property of a [[Boolf-term#TT|truth table]] with a specific length.


the term ''noble'' is applied to [[Boolf-term#TT|truth tables]] rather than [[Boolf-term#BF|Boolean functions]].&lt;/small&gt;

misleading to apply the term noble to Boolean functions. It really is a property of a truth table with a specific length.
{{Zhegalkin matrix/Triangle Pi|collapsed}}

{{Zhegalkin matrix/Triangle Phi|collapsed}}

{{Collapsible START|3-ary nobles assigned to vertices of a tesseract|open gap-below}}
[[File:3-ary nobles in tesseract.svg|500px]]
{{Collapsible END}}

{{Collapsible START|illustration of fixed points &lt;math&gt;\Phi_3&lt;/math&gt; in permutation &lt;math&gt;\Pi_3&lt;/math&gt;|collapsed gap-below}}
[[File:Zhegalkin 256; fixed.svg|1320px]]
{{Collapsible END}}

{{Collapsible START|&lt;math&gt;\Phi_3&lt;/math&gt; as 8×16 matrix {{spaces|8}} &lt;math&gt;\Phi_4&lt;/math&gt; as 16×256 matrix|collapsed  gap-below}}
[[File:Fixed points in Zhegalkin permutation 3.svg|thumb|left|195px|row 3]]
{{clear}}
[[File:Fixed points in Zhegalkin permutation 4.svg|thumb|left|1440px|row 4]]
{{clear}}
The images show how &lt;math&gt;\Phi_n&lt;/math&gt; is derived from &lt;math&gt;\Pi_{n-1}&lt;/math&gt;, the permutation of the same length.&lt;br&gt;
The long matrices have two halves:&lt;br&gt;
In the upper half the bit pattern in column &lt;math&gt;k&lt;/math&gt; is that of &lt;math&gt;\Xi_{n-1,~k} ~ = ~ \Pi_{n-1,~k} \oplus k&lt;/math&gt;.
&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;small&gt;(Compare triangle Ξ in the next section.)&lt;/small&gt;&lt;br&gt;
The bit pattern in the lower half is identical to that of the gray column indices.

&lt;math&gt;\Phi_{n,~k} ~~ = ~~ \Xi_{n-1,~k} + \bigl(2^{2^{n-1}} \cdot k \bigr) ~~ = ~~ \bigl(\Pi_{n-1,~k} \oplus k \bigr) + \bigl(2^{2^{n-1}} \cdot k \bigr)&lt;/math&gt;

The small matrices on the left are divided in the same way:&lt;br&gt;
The upper half is a Sierpiński triangle without the main diagonal, and the lower half is the main diagonal.

These properties of noble Boolean functions can be derived from this:&lt;br&gt;
* They are even, i.e. place 0 is always false.
* The 1-bit places (e.g. 1, 2, 4, 8) have the same truth value. &lt;small&gt;(Those where it is false/true shall be called weak/strong.)&lt;/small&gt;
* Half of them are evil/odious, which is indicated by the last place being false/true. &lt;small&gt;(The odious ones are on the right, just like in triangle &amp;Pi;.)&lt;/small&gt;
{{Collapsible END}}

{{Noble Boolean functions/row sums}}

===quadrants===

It is easily seen, that the left and right half of each row differ only in the last digit.&lt;br&gt;
Those on the left/right have even/odd weight. They shall be called evil/odious.

{{Noble Boolean functions/Python half rows}}

There is a second way to partition the nobles in two halves:&lt;br&gt;
Those in even/odd places of the triangle row have false/true entries in all 1-bit places of their truth table. They shall be called weak/strong.

So the nobles can be partitioned into four quadrants by depravity and strength.

{{Collapsible START|illustration of quadrants for ''n'' &amp;equals; 3|open}}
Vertically adjacent quadrants contain relative complements. Horizontally adjacent quadrants differ only in the central vertex.&lt;br&gt;
&lt;small&gt;(40 and 214 are relative complements. 40 and 168 differ only in the central vertex.)&lt;/small&gt;&lt;br&gt;

The 16 noble 3-ary Boolean functions form 8 factions. So there are 2 royal factions.
[[File:Noble 3-ary Boolean functions.svg|650px|center]]
{{Collapsible END}}

Nobles that are evil and weak shall be called '''''royal'''''.&lt;br&gt;
Each noble corresponds to a royal, and can easily be derived from it. &lt;small style=&quot;opacity: .5;&quot;&gt;(A royal corresponds to itself.)&lt;/small&gt;&lt;br&gt;
When a Boolean function is royal, its whole [[Boolf-EC#P|faction]] is royal.&lt;br&gt;
The function with the smallest Zhegalkin index in a faction shall be called '''''king''''', and be used to represent it.&lt;br&gt;
So all nobles of a given arity can effectively be represented by a rather short list of kings.

{{Noble Boolean functions/royal triangle|collapsed}}

{{Collapsible START|4-ary royals in 16×64 matrix|collapsed light gap-below}}
The columns of this matrix are the 4-ary royal Boolean functions.&lt;br&gt;
[[File:4-ary royal Boolean functions in matrix.svg|x180px]]&lt;br&gt;
Compare &lt;math&gt;\Phi_4&lt;/math&gt; as 16×256 matrix, shown above.&lt;br&gt;
&lt;small&gt;The 16×6 matrix on the left is shown as the 16×8 matrix from that file, with the left and right column blacked out.&lt;/small&gt;
{{Collapsible END}}

{{Noble Boolean functions/triangle of kings|collapsed}}

{{Collapsible START|representatives of 4-ary noble factions|collapsed}}
[[File:Representatives of 4-ary noble factions.svg|1200px]]

The top left corner in each 2×2 matrix is a king. The other corners are its equivalents in the other quadrants.&lt;br&gt;
&lt;small style=&quot;opacity: .5;&quot;&gt;(Vertically adjacent quadrants contain relative complements. Horizontally adjacent quadrants differ only in the central vertex.)&lt;/small&gt;

The two tables below correspond to the image above. The one on the left shows the same numbers as the image.&lt;br&gt;
The one on the right shows the [[smallest Zhegalkin index]] for each of the 44 factions. &lt;small style=&quot;opacity: .5;&quot;&gt;(They differ only for the odd factions, i.e. those with green and blue background.)&lt;/small&gt;

{| style=&quot;width: 100%;&quot;
| {{Noble reps/outer/match}}
| {{Noble reps/outer/minimal}}
|}

The black integers are Zhegalkin indices. The gray numbers below are their noble indices, i.e. the positions in the sequence of nobles.&lt;br&gt;
The beige numbers in the image are faction sizes, i.e. the number of different permutations of the example shown.
{{Collapsible END}}

{{Collapsible START|4-ary kings|collapsed light gap-above}}
[[File:Representatives of 4-ary royal factions.svg|800px]]
{{Collapsible END}}

===group under exclusive or===

With XOR as a group operation the ''n''-ary noble and royal Boolean functions form a power of the {{w|cyclic group}} C&lt;sub&gt;2&lt;/sub&gt;.

{{Collapsible START|Python example|collapsed light gap-below}}
The 3-ary nobles form the group C&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;4&lt;/sup&gt;.

The Python operator &lt;code&gt;^&lt;/code&gt; represents the {{w|bitwise operation#XOR|bitwise XOR}}.

&lt;source lang=&quot;python&quot;&gt;
nobles = [0,  30,  40, 54,  72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]

for i, a in enumerate(nobles):
    for j, b in enumerate(nobles):
        assert nobles.index(a ^ b) == i ^ j
&lt;/source&gt;

This works for any row of the noble triangle &lt;math&gt;\Phi&lt;/math&gt;, or from the royal triangle. &lt;small&gt;(But not from the triangle of kings.)&lt;/small&gt;
{{Collapsible END}}


==patrons==

The XOR of twins is noble, i.e. the XOR of a key and a value in &lt;math&gt;\Pi_n&lt;/math&gt; is an entry of &lt;math&gt;\Phi_n&lt;/math&gt;.&lt;br&gt;
For a Boolean function this means, that the XOR of its ANF and its truth table is a noble Boolean function, which shall be called its '''''patron'''''.&lt;br&gt;
The patron of a noble is the contradiction.

{{Zhegalkin matrix/triangle Xi|collapsed}}

For &lt;math&gt;n \ge 1&lt;/math&gt; the entries in &lt;math&gt;\Xi_n&lt;/math&gt; are repetitions of those in &lt;math&gt;\Phi_n&lt;/math&gt;. &lt;span style=&quot;opacity: .6;&quot;&gt;E.g. &lt;math&gt;\Xi_2&lt;/math&gt; contains the entries &lt;math&gt;\{0, 6, 8, 14\}&lt;/math&gt;, each repeated four times.&lt;/span&gt;

The set of places where &lt;math&gt;\Xi_n&lt;/math&gt; has entries &lt;math&gt;\Phi_{n,~k}&lt;/math&gt; can be calculated by XORing &lt;math&gt;\Pi_{n-1,~k}&lt;/math&gt; with the entries of &lt;math&gt;\Phi_n&lt;/math&gt;.

===3-ary Boolean functions with patron 168===

{{Zhegalkin matrix/matrix pi xor phi}}

{{Zhegalkin matrix/clusters with patron 168}}

{{Collapsible START|positions in matrix|collapsed gap-above}}
[[File:3-ary Boolean functions; patrons 10.svg|thumb|center|500px|Boolean functions with patron 168 in octeract matrix&lt;br&gt;&lt;small&gt;(This is the transpose of [[c:File:3-ary Boolean functions; quaestor 03 (indices).svg|Zhegalkin indices with quaestor 3]].)&lt;/small&gt;]]
{{Collapsible END}}


[[Category:Boolean functions; Zhegalkin stuff]]</text>
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      <text bytes="381" xml:space="preserve">== Summary ==
{{Information
|Description= Peer review of [[WikiJournal Preprints/Globally Popular Pet Reptile Leopard Gecko (Eublepharis macularius) Demonstrates Capability in Using Running Wheel Voluntarily – Is It Locomotion Play?]]
|Source= WikiJournal of Science
|Date= 2023-11-13
|Author= Wolf Huetteroth
|Permission=
}}

==Licensing==
{{cc-by-4.0}}
[[Category:WikiJournal]]</text>
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{{Information
|Description= English: Peer review comments for [[WikiJournal Preprints/Design effect]]
|Source= [[WikiJournal of Science]]
|Date= 3/1/2024
|Author= Charles DiSogra
|Permission=
}}

==Licensing==
{{cc-by-4.0}}
[[Category:WikiJournal]]</text>
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{{Information
|Description= English: Peer review comments for [[WikiJournal Preprints/Bioclogging]]
|Source=[[WikiJournal of Science]]
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|Author= Anonymous (submitted in confidence)
|Permission=
}}

==Licensing==
{{cc-by-4.0}}
[[Category:WikiJournal]]</text>
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{{Information
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    <title>User:Guy vandegrift/sandbox/BLANK</title>
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      <comment>/* Table */</comment>
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      <text bytes="666" xml:space="preserve">==Surreal numbers table==
Rescued from first attempt (and discarded by me)
{| class=&quot;wikitable floatleft&quot;
|+
!Day
!#
!Set
|-
|0
|0
|&lt;nowiki&gt;{|}&lt;/nowiki&gt;
|-
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| -1
|&lt;nowiki&gt;{|0}&lt;/nowiki&gt;
|-
|1
|&lt;nowiki&gt;{0|}&lt;/nowiki&gt;
|-
| rowspan=&quot;4&quot; |2
| -2
|&lt;nowiki&gt;{|-1}&lt;/nowiki&gt;
|-
| -1/2
|&lt;nowiki&gt;{-1|0}&lt;/nowiki&gt;
|-
|1/2
|&lt;nowiki&gt;{0|1}&lt;/nowiki&gt;
|-
|2
|&lt;nowiki&gt;{1|}&lt;/nowiki&gt;
|-
| rowspan=&quot;8&quot; |3
| -3
|&lt;nowiki&gt;{|-2}&lt;/nowiki&gt;
|-
| -3/2
|&lt;nowiki&gt;{-2|-1}&lt;/nowiki&gt;
|-
| -3/4
|&lt;nowiki&gt;{-1|-1/2}&lt;/nowiki&gt;
|-
| -1/4
|&lt;nowiki&gt;{-1/2|0}&lt;/nowiki&gt;
|-
|1/4
|&lt;nowiki&gt;{0|1/2}&lt;/nowiki&gt;
|-
|3/4
|&lt;nowiki&gt;{1/2|1}&lt;/nowiki&gt;
|-
|3/2
|&lt;nowiki&gt;{1|2}&lt;/nowiki&gt;
|-
|3
|&lt;nowiki&gt;{2|}&lt;/nowiki&gt;
|}</text>
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      <text bytes="12" xml:space="preserve">==Universe==</text>
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[[File:Hasse_diagram_of_powerset_of_3.svg|thumb]]</text>
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[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|upright=1.2]]</text>
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[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|inverted=1.2]]</text>
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[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|upright=1.2]]</text>
      <sha1>6hcns7x8eajgn52d3x2u21ljh9xpw3c</sha1>
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    <revision>
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      <text bytes="213" xml:space="preserve">==Universe==
[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|upright=1.2]]   Naive set theory is a way of talking about sets (collections of objects) using plain language, without the strict rules of formal logic.</text>
      <sha1>ts7orqwd065in4zz0sf8mvvzp9mecob</sha1>
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      <text bytes="518" xml:space="preserve">==Universe==
[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|upright=1.2]]   [[w:Naive set theory]] is a way of talking about sets (collections of objects) using plain language, without the strict rules of formal logic.  A good example of the success of naive set theory is counting the number of possible sets, given the proposition that the &quot;universe&quot; consists of sets that are only allowed to contain three letters: x, y, and z. Even this &quot;universe&quot; allows us to create the eight different sets shown in the figure.</text>
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      <text bytes="568" xml:space="preserve">==Universe==
[[File:Hasse_diagram_of_powerset_of_3.svg|thumb|upright=1.2]]   [[w:Naive set theory|Naive set theory]] is a way of talking about sets (collections of objects) using plain language, without the strict rules of formal logic.  A good example of the success of naive set theory is counting the number of possible sets, given the proposition that the &quot;universe&quot; consists of sets that are only allowed to contain three letters: x, y, and z. Even this &quot;universe&quot; allows us to create the eight different sets shown in the figure.

[[w:Peano axioms|Peano axioms]]</text>
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{{Response to review
 |Author   = Richard Digirolamo
 |link       = https://en.wikiversity.org/w/index.php?title=WikiJournal_Preprints/Globally_Popular_Pet_Reptile_Leopard_Gecko_(Eublepharis_macularius)_Demonstrates_Capability_in_Using_Running_Wheel_Voluntarily_%E2%80%93_Is_It_Locomotion_Play%3F&amp;oldid=2595874
 |date       = 11 March 2024
 |pdf        = Locomotor_play_in_leopard_gecko_-_Wolf_Huetteroth_RD_responses-31-.pdf
 |text       =
See uploaded file
}}

== Licensing ==
{{self|GFDL|cc-by-4.0}}
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      <text bytes="482" xml:space="preserve">== Summary ==
{{Information
|Description=Response to the [[:File:Locomotor play in leopard gecko - Wolf Huetteroth.pdf|peer review by Wolf Huetteroth]] of [[WikiJournal Preprints/Globally Popular Pet Reptile Leopard Gecko (Eublepharis macularius) Demonstrates Capability in Using Running Wheel Voluntarily – Is It Locomotion Play?]]
|Source= WikiJournal of Science
|Date= 2024-03-02
|Author= Richard Digirolamo
|Permission=
}}

==Licensing==
{{cc-by-4.0}}
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      <text bytes="499" xml:space="preserve">== Summary ==
{{Information
|Description=Author's response to [[:File:Locomotor play in leopard gecko - Second Reviewer comments.pdf|the peer review by second reviewer]] of [[WikiJournal Preprints/Globally Popular Pet Reptile Leopard Gecko (Eublepharis macularius) Demonstrates Capability in Using Running Wheel Voluntarily – Is It Locomotion Play?]]
|Source= WikiJournal of Science
|Date= 2024-03-02
|Author=Richard Digirolamo
|Permission=
}}

==Licensing==
{{cc-by-4.0}}
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  <page>
    <title>User:Dc.samizdat/A symmetrical arrangement of 120 11-cells</title>
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      <text bytes="139290" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 121-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, and that their directionality matters; he concluded that since the means ''determine'' the ends, they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 121-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 121-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (121-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
      <sha1>8jn2xpj9nylbg3b59uuofr8q0z2bgd3</sha1>
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      <text bytes="139631" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 121-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 121-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 121-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (121-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="139654" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 121-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 121-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 121-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (121-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="139653" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 121-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 121-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 121-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (121-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
      <sha1>qhvb0kaf0d33ft9gocqosp30wux0rpn</sha1>
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    <revision>
      <id>2624842</id>
      <parentid>2624771</parentid>
      <timestamp>2024-05-02T21:05:23Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <comment>137-cell</comment>
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      <text bytes="139653" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
      <sha1>f93c949rpawvnpf2aws3bdm4n4sip1h</sha1>
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    <revision>
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      <timestamp>2024-05-02T21:09:44Z</timestamp>
      <contributor>
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      <comment>/* The rings of the 11-cells */</comment>
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      <text bytes="139665" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads, 100 hexads, 120 heptads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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    <revision>
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      <timestamp>2024-05-02T21:13:28Z</timestamp>
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      <comment>/* The rings of the 11-cells */</comment>
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      <text bytes="139667" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads&lt;br&gt;= 120 heptads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="139650" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold decagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold polygonal symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
      <sha1>9mmghd12pnm4y81l87uyn1s3zdk399v</sha1>
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    <revision>
      <id>2624847</id>
      <parentid>2624845</parentid>
      <timestamp>2024-05-02T21:50:17Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* The rings of the 11-cells */</comment>
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      <text bytes="140084" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}⋎(2-𝝓)&lt;/math&gt; great hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="140196" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlyand(2-𝝓)&lt;/math&gt; great hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="140198" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-𝝓)&lt;/math&gt; great hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="140198" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; great hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="140192" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="140194" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 regular 5-cells in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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    <revision>
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      <timestamp>2024-05-02T22:10:53Z</timestamp>
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      <comment>/* The 137-cell regular convex 4-polytope */</comment>
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      <text bytes="140187" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|11 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|11
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|11
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
      <sha1>pw7gp6o7bm7w4ueyxj15uodi9anui3w</sha1>
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      <timestamp>2024-05-02T23:54:10Z</timestamp>
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      <comment>/* The 137-cell regular convex 4-polytope */</comment>
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      <text bytes="140194" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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    <revision>
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      <timestamp>2024-05-03T04:32:09Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* The perfection of Fuller's cyclic design */</comment>
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      <text bytes="142086" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|
|-
!
!
!
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|rowspan=2|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="142100" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|
|-
!
!
!
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|
|-
!
!
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!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|rowspan=2|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142144" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
|-
!colspan=5|
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|rowspan=2|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="142149" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
|-
!colspan=5|title
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
|-
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|rowspan=2|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142174" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
|-
!colspan=5|title
|-
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!header
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|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|
|rowspan=2|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|rowspan=2|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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      <text bytes="142171" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
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|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
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|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
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|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
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|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
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|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
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|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
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|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142154" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
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!header
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|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
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|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
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!footer
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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      <text bytes="142158" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell_t0_F4.svg|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
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!header
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|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
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|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142448" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!footer
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!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="142448" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142452" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
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!footer
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142444" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: pink;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: pink;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!footer
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!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="142450" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: #FFF5FF;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: #FFF5FF;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142450" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
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!header
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!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: hue:300;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: hue:300;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
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!footer
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
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      <text bytes="142450" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: #FF40FF;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: hue:300;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!footer
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!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="142454" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: seashell;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: palegreen;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: palegreen;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
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== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142453" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=5|title
|-
!
!
!header
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
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!footer
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|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142476" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexad
!pentad
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|
|
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!footer
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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      <text bytes="142462" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexad
!pentad
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|
|
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexad
|pentad
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11-point (11-cell)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|
|
|[[File:5-cell.gif|100px]]&lt;br&gt;137-point (137-cell)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142524" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="142588" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142588" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;24-point (24-cell)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;120-point (600-cell)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;8-point (16-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;5-point (5-cell)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142609" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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      <text bytes="142673" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;5 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142676" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
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      <text bytes="142738" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|[[Rhombicosidodecahedron.gif|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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      <text bytes="142738" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[Rhombicosidodecahedron.gif|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="142743" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Rhombicosidodecahedron.gif|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="142795" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="143074" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="143197" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="143235" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here, but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="143675" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here, but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="143750" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Pieski|3 Mar 2018|loc=nonuniform rhombicosidodecahedron}} but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2 Apr 2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="143747" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Piesk rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Piesk rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="143748" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Piesk rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller pentagons and narrower rhomboids. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Cite journal|last1=Schleimer|first1=Saul|last2=Segerman|first2=Henry|date=2013|title=Puzzling the 120-cell|journal=Notices Amer. Math. Soc.|volume=62|issue=11|pages=1309–1316|doi=10.1090/noti1297 |arxiv=1310.3549 |s2cid=117636740 }}
*{{citation
 | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin
 | last2 = Hamlin | first2 = James F.
 | contribution = The Regular 4-dimensional 57-cell
 | doi = 10.1145/1278780.1278784
 | location = New York, NY, USA
 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 }}
* {{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|publisher=Houghton Mifflin|place=Boston}}
{{Refend}}</text>
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      <text bytes="143745" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Piesk rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller pentagons and narrower rhombs. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|15 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|40
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|11
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|15.5°
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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{{Refend}}</text>
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      <text bytes="143742" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the quasi-regular 4-polytope {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells. Eleven 11-cells (plural) are the 137-cell regular convex 4-polytope {3, 5, 3}.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The child must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cube]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The cell of the 11-cell is an abstract 6-point hemi-icosahedron containing 5 regular 5-cells, handsomely illustrated by Séquin.{{Sfn|Séquin|Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing concentric rings of pentad polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|120-point (600-cell)]] faces, separated from each other by rectangles.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]], a rectified rhombic triacontahedron similar to the one from the catalog shown here,{{Sfn|Piesk rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller pentagons and narrower rhombs. The red pentagons are 120-cell faces. The 120-cell contains 120 of these 11-cell cells.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
Moxness's 60-point (hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|Petrie|1938|p=4|loc=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad{{Sfn|Steinbach|1997|loc=Golden fields: A case for the Heptagon|pages=22–31}}
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes. The pentad is a 5-point (regular 5-cell), the hexad is half a 12-point (16-cell), and the heptad is half an 11-point (11-cell).

Each regular 5-cell in the 120-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies in the 120-cell completely orthogonal to another pentad building block. They are two completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. The hexad is a quasi-regular polyhedron that also has {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairs of pentads and their completely orthogonal hexads, and there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct an 11-point ([[W:11-cell|11-cell]]) 4-polytope, which magically contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has its origin in three equivalent root systems. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything else can be constructed by compounding hexads, and we shall see that everything as large as the 120-cell also has a construction from heptads which are a product of a pentad and a hexad. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes related by an equals sign between them give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics, by [[W:Noether's theorem|Noether's theorem]].

=== From pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad truncated cuboctahedron), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space takes on depends on the point at which it is embedded. A 3-polytope found inscribed in a 4-polytope concentric to their common center acts like another 4-polytope, even if it resembles a polyhedron that is not usually seen as a 4-polytope (e.g. it may have fewer than 5 vertices). A 3-polytope found concentric to an off-center point in the 4-polytope's interior acts like a cell. A 3-polytope found concentric to a point on the surface of a 4-polytope acts as a vertex figure. All 3-polytopes embedded in 4-space may be described both as a spherical 3-polytope and as a flat 4-polytope; e.g. a vertex figure can be described as a spherical 3-polytope and as a 4-pyramid with the 3-polytope as its base.}} Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (quadrad regular tetrahedra and hexad truncated cuboctahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 out of 120 completely disjoint instances of a 4-polytope. The hexad cells are 6 out of 200 never-disjoint instances of a 3-polytope. Each 11-cell shares each of its hexad cells with 3 other 11-cells, and each of the 600 vertices occurs in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (pentad) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (pentad) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== 11 of them ===
11-cells and their heptad build block are not required (or permitted) in any of the regular convex 4-polytopes up to and including the 600-cell. 11-cells and their heptad building blocks are present and required in regular convex 4-polytopes larger than the 600-cell.

There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of 11 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum 11 of them. That is, there is only a single Hopf fibration of the 11 great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 11 regular 5-cells and 10 other instances of itself. 

== The rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|2010|loc=Goucher's ''[[W:120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[W:120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious candidate to be the 11-cell's Hopf map is Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other candidate. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|loc=''
Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexad symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section two sections. 

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section.

== The 137-cell regular convex 4-polytope ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

=== ... ===
{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=6|title
|-
!
!
!hexads
!pentads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:5-cell.gif|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!hexads
|pentads
!
!
|}

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022|loc=[[W:Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=[[W:SO(4)|SO(4)]]}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies completely orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its completely orthogonal hexad's planes (wz, wx, wy).{{Efn|name=six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed five 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosahedron). The five 12-points were inscribed in the 60-point such that their corresponding vertices were close together, the 5 vertices of a pentagon face, and the 12 little pentagon faces were joined to their opposite pentagon faces by 5 reflex edges of 5 different Jessen's. The contraction of those 5 vertices into 1, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing them. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (hemi-icosahedron), the 11-cell's abstract hexad cell. From this we learned that the hexad's situation is that it occurs in bundles of 5, close together in the sense of adjacent, like the fiber bundles of great circle rings in a Hopf fibration. 

To summarize, the 12-point (hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell).  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederon&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== The 12-point Legendre vertex binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre 3-polytope) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cube), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubes), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-taliesins ===
The 11-cells are the ''taliesin'' 4-polytopes. 

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in flat [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].

...

=== The ''n''-leviathon ===


{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!Chord
!Arc
!colspan=2|L√1
!3-sphere
!Vertex
|- style=&quot;background: seashell;&quot;|
|#1 △
|15.5~°
|{{radic|0.𝜀}}{{Efn|name=fractional square roots|group=}}
|0.270~
|rowspan=30|[[File:15 major chords.png|500px|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the regular or quasi-regular convex 4-polytopes or their characteristic great circle rings. There are 30 distinct 4-space chordal distances between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to name any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]]
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: seashell;&quot;|
|#2 &lt;big&gt;☐&lt;/big&gt;
|25.2~°
|{{radic|0.19~}}
|0.437~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: yellow;&quot;|
|#3 &lt;big&gt;✩&lt;/big&gt;
|36°
|{{radic|0.𝚫}}
|0.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#4 △
|44.5~°
|{{radic|0.57~}}
|0.757~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#5 △
|60°
|{{radic|1}}
|1
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: yellow;&quot;|
|#6 &lt;big&gt;✩&lt;/big&gt;
|72°
|{{radic|1.𝚫}}
|1.175~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#7 &lt;big&gt;☐&lt;/big&gt;
|90°
|{{radic|2}}
|1.414~
|{{blue|&lt;big&gt;'''54'''&lt;/big&gt;}} 9{3,4}
|- style=&quot;background: seashell;&quot;|
| #8 △
|104.5~°
|{{radic|2.5}}
|1.581~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#9 &lt;big&gt;✩&lt;/big&gt;
|108°
|{{radic|2.𝚽}}
|1.618~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: paleturquoise;&quot;|
|#10 △
|120°
|{{radic|3}}
|1.732~
|{{blue|&lt;big&gt;'''32'''&lt;/big&gt;}} 4{4,3}
|- style=&quot;background: seashell;&quot;|
|#11 △
|135.5~°
|{{radic|3.43~}}
|1.851~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: yellow;&quot;|
|#12 &lt;big&gt;✩&lt;/big&gt;
|144°
|{{radic|3.𝚽}}
|1.902~
|{{blue|&lt;big&gt;'''24'''&lt;/big&gt;}} 2{3,5}
|- style=&quot;background: seashell;&quot;|
|#13 &lt;big&gt;☐&lt;/big&gt;
|154.8~°
|{{radic|3.81~}}
|1.952~
|{{blue|&lt;big&gt;'''12'''&lt;/big&gt;}} 2{3,4}
|- style=&quot;background: seashell;&quot;|
|#14 △
|164.5~°
|{{radic|3.93~}}
|1.982~
|{{blue|&lt;big&gt;'''4'''&lt;/big&gt;}} {3,3}
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{blue|&lt;big&gt;'''1'''&lt;/big&gt;}} &lt;br&gt;
|}

....my fan of major chords circle diagram, with left side and right side legends that are the leftmost columns (give #, arc, both √2 and √1 radius lengths, formula only if noteworthy e.g.  #5 = ''r'' and #9 = 𝝓''r'') and rightmost column of the major chords table.....

...say the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....ask what's with that crooked #11 chord?

...abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article...


...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[w:120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[w:600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[w:24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of &lt;math&gt;11^2&lt;/math&gt; of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were colleagues of their contemporaries the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not a singleton convex 4-polytope, not a honeycomb,{{Sfn|Coxeter|1970|loc=Twisted Honeycombs}} and not an [[W:abstract polytope|abstract 4-polytope]]. The 11-point (11-cell) has a concrete quasi-regular realization {{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {3, 5, 3} as the 137-point (137-cell) regular convex 4-polytope. There are seven regular convex 4-polytopes. 

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=Clifford Spinors and Root System Induction: H4 and the Grand Antiprism}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Play with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Jung|1961|loc=Carl Jung}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1967|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
      <sha1>m5amojq1gdg6wvs4az36q6rumxmvun2</sha1>
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  </page>
  <page>
    <title>User:Silas Andzie</title>
    <ns>2</ns>
    <id>304342</id>
    <revision>
      <id>2624588</id>
      <parentid>2624576</parentid>
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      <contributor>
        <username>Silas Andzie</username>
        <id>2970181</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3779" xml:space="preserve">Globe Education Centre-Travelling Abroad furthering studies.
Sunny FM-Direction to outfit
1Gempa Legon 
2Toyota Ghana UG opposite ask No.4
0244783000-0548393600
Return Month to call April back

GHANA EDUCATION SERVICE 

GETFUND 

OFFICIAL NUMBER:
0544323020

OFFICIAL WEBSITE
scholarship.get fund.gov.gh
------------------------------

 METROPOLITAN UNIVERSITY 
   COLLEGE OF MEDICINE

ADMISSION LETTER

SCHOLARSHIP $40%

CONTACT:+1(1813)-755-6705-
+1(268)562-9262-+(268)714-4680
info@muantigua.org
admissions@muantigua.org
Sir.Ryan's building,Newgate street st.John's Antigua,Antigua and Bartuda

FULLNAME:SILAS KWAMENA ANDZIE

EMAIL:Sandzie79@yahoo.|Silasandzie@gmail.com

PHONE NUMBER:GH+233 0249649678

DATE OF BIRTH :17-03-1979| 08-19-1987

WHICH COUNTRY ARE YOU FROM? GHANA-WESTAFRICA.

WHAT IS YOUR EDUCATIONAL BACKGROUND OR LEVEL? UNIVERSITY

PLEASE ENTER YOUR PRIMARY CONTACT NUMBER WITH COUNTRY CODE.GH+2330249649678

A Curriculum Vitae (C.V)
1) Personal Data: A Student of Ghana Christian University College.Advancing excellence in transforming leadership (Amrahia-Dodowa Rd.)
(Accredited by National Board of The Ministry Of Education)
Advancing Excellence in Transformational Leadership (Since 1966)

2)Surname:Andzie First name:Silas other name:Kwamena
3)Age:45 
4)Sex:Male
5)Qualification And Certificate of Appreciation
And Transcript:
a)A.G.P.A.:3.03  Enrole School Of Theology And Ministry 
Bachalor of Degree.Ghana Christian University College

b)Transfer to School Of Medicine Bachalor Of Degree.Metropolitan University College Of Medicine

c) Awarded Certificate Of Appreciation dedicated service as the Chaplain of Ghana Christian University College.
Students'Representative Council.(2006-2008)

d)Awarded Certificate Of Attendance &quot;Preparing Apostles for the Apostolic Season&quot;Annual Congress November 2006 by Abundant Life Ministries International.

e) Awards Certificate of Participation Holy Spirit Conference&quot;Restoration Of True Pastors by the Mystery Of the Body Of Christ Ministry.(October 2004)

f)Awarded Certificate Of Parcipation by Ghana Christian University College.&quot;Strategic Planning&quot; May 2007.

g)Awarded Certificate Of Participation &quot;Restoring True Spiritual Fatherhood&quot;Leadership Conference April 2006 by World Changers International.

h)Awarded Certificate of Bible Correspondence Course the Gospel Of St.John 96.84% Average Marks 48.42 by &quot;The Gospel Messagers&quot;Ministry.

5) Working
Experience:

i)Certificate of Incorporation
 Rescued Community Church             
   International issued for  service December 2014 and 

ii)Certificate To Commence
     Business 
Rescued Community Church International issued for service June 2015.

6)Chief Internal a)Auditor:Adjei-Nimako&amp;Associates
 b)Finance,Marketing Management Consultants
c) Secretary details:
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References:Rev.Silas Andzie,Founder &amp; President  of Rescued Community Church International and the support Ministries Department Current Account.Access Bank.

Mr.Godfred Yaw Oto.Korle-bu Heart Cardio Centre and Copy Media Printers Manager.

Pastor Augustine Kwaku Adjei.Associate Head Branch Redemption Hour Faith Ministries and Electronic Manager.

RESCUED COMMUNITY CHURCH INTERNATIONAL

Dear Sir|Madam,
LETTER OF RECOMMENDATION APPROVAL FOR LOCAL SPONSORSHIP TO DO STUDY ABROAD

The above stated company's name with the HeadPastor politely write to recommend approval for sponsorship acceptance letter to proceed study abroad.
  
However,through critical observation search made by the admission letters submitted to both schools.Namely,Ghana Christian University College and the Metropolitan University College Of Medicine and the curriculum vitae (CV)is witnessing and proofing the evidence account that deserve to be recommended to carryout this carrier goal</text>
      <sha1>8krnli0mlzg4i4obwrmbvh24i64q3d3</sha1>
    </revision>
    <revision>
      <id>2624590</id>
      <parentid>2624588</parentid>
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      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4015" xml:space="preserve">Globe Education Centre-Travelling Abroad furthering studies.
Sunny FM-Direction to outfit
1Gempa Legon 
2Toyota Ghana UG opposite ask No.4
0244783000-0548393600
Return Month to call April back

GHANA EDUCATION SERVICE 

GETFUND 

OFFICIAL NUMBER:
0544323020

OFFICIAL WEBSITE
scholarship.get fund.gov.gh
------------------------------

 METROPOLITAN UNIVERSITY 
   COLLEGE OF MEDICINE

ADMISSION LETTER

SCHOLARSHIP $40%

CONTACT:+1(1813)-755-6705-
+1(268)562-9262-+(268)714-4680
info@muantigua.org
admissions@muantigua.org
Sir.Ryan's building,Newgate street st.John's Antigua,Antigua and Bartuda

FULLNAME:SILAS KWAMENA ANDZIE

EMAIL:Sandzie79@yahoo.|Silasandzie@gmail.com

PHONE NUMBER:GH+233 0249649678

DATE OF BIRTH :17-03-1979| 08-19-1987

WHICH COUNTRY ARE YOU FROM? GHANA-WESTAFRICA.

WHAT IS YOUR EDUCATIONAL BACKGROUND OR LEVEL? UNIVERSITY

PLEASE ENTER YOUR PRIMARY CONTACT NUMBER WITH COUNTRY CODE.GH+2330249649678

A Curriculum Vitae (C.V)
1) Personal Data: A Student of Ghana Christian University College.Advancing excellence in transforming leadership (Amrahia-Dodowa Rd.)
(Accredited by National Board of The Ministry Of Education)
Advancing Excellence in Transformational Leadership (Since 1966)

2)Surname:Andzie First name:Silas other name:Kwamena
3)Age:45 
4)Sex:Male
5)Qualification And Certificate of Appreciation
And Transcript:
a)A.G.P.A.:3.03  Enrole School Of Theology And Ministry 
Bachalor of Degree.Ghana Christian University College

b)Transfer to School Of Medicine Bachalor Of Degree.Metropolitan University College Of Medicine

c) Awarded Certificate Of Appreciation dedicated service as the Chaplain of Ghana Christian University College.
Students'Representative Council.(2006-2008)

d)Awarded Certificate Of Attendance &quot;Preparing Apostles for the Apostolic Season&quot;Annual Congress November 2006 by Abundant Life Ministries International.

e) Awards Certificate of Participation Holy Spirit Conference&quot;Restoration Of True Pastors by the Mystery Of the Body Of Christ Ministry.(October 2004)

f)Awarded Certificate Of Parcipation by Ghana Christian University College.&quot;Strategic Planning&quot; May 2007.

g)Awarded Certificate Of Participation &quot;Restoring True Spiritual Fatherhood&quot;Leadership Conference April 2006 by World Changers International.

h)Awarded Certificate of Bible Correspondence Course the Gospel Of St.John 96.84% Average Marks 48.42 by &quot;The Gospel Messagers&quot;Ministry.

5) Working
Experience:

i)Certificate of Incorporation
 Rescued Community Church             
   International issued for  service December 2014 and 

ii)Certificate To Commence
     Business 
Rescued Community Church International issued for service June 2015.

6)Chief Internal a)Auditor:Adjei-Nimako&amp;Associates
 b)Finance,Marketing Management Consultants
c) Secretary details:
Silas Kwamena Andzie

References:Rev.Silas Andzie,Founder &amp; President  of Rescued Community Church International and the support Ministries Department Current Account.Access Bank.

Mr.Godfred Yaw Oto.Korle-bu Heart Cardio Centre and Copy Media Printers Manager.

Pastor Augustine Kwaku Adjei.Associate Head Branch Redemption Hour Faith Ministries and Electronic Manager.

RESCUED COMMUNITY CHURCH INTERNATIONAL

Dear Sir|Madam,
LETTER OF RECOMMENDATION APPROVAL FOR LOCAL SPONSORSHIP TO DO STUDY ABROAD AND MINISTRY

The above stated company's name with the HeadPastor politely write to recommend approval for sponsorship acceptance letter to proceed study abroad.
  
However,through critical observation search made by the admission letters submitted to both schools.Namely,Ghana Christian University College and the Metropolitan University College Of Medicine and the curriculum vitae (CV)is witnessing and proofing the evidence account that deserve to be recommended to carryout this carrier goal in Medicine and Ministry.

We will therefore be glad to hear from the administrative body of Ghana National Petroleum Corporation to accept this offer in order,to pay away to get local sponsorship to proceed study outside.</text>
      <sha1>bm4bmnvsi56kuohkd674iteuza1401c</sha1>
    </revision>
    <revision>
      <id>2624593</id>
      <parentid>2624590</parentid>
      <timestamp>2024-05-02T13:40:57Z</timestamp>
      <contributor>
        <username>Silas Andzie</username>
        <id>2970181</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4030" xml:space="preserve">Globe Education Centre-Travelling Abroad furthering studies.
Sunny FM-Direction to outfit
1Gempa Legon 
2Toyota Ghana UG opposite ask No.4
0244783000-0548393600
Return Month to call April back

GHANA EDUCATION SERVICE 

GETFUND 

OFFICIAL NUMBER:
0544323020

OFFICIAL WEBSITE
scholarship.get fund.gov.gh
------------------------------

 METROPOLITAN UNIVERSITY 
   COLLEGE OF MEDICINE

ADMISSION LETTER

SCHOLARSHIP $40%

CONTACT:+1(1813)-755-6705-
+1(268)562-9262-+(268)714-4680
info@muantigua.org
admissions@muantigua.org
Sir.Ryan's building,Newgate street st.John's Antigua,Antigua and Bartuda

FULLNAME:SILAS KWAMENA ANDZIE

EMAIL:Sandzie79@yahoo.|Silasandzie@gmail.com

PHONE NUMBER:GH+233 0249649678

DATE OF BIRTH :17-03-1979| 08-19-1987

WHICH COUNTRY ARE YOU FROM? GHANA-WESTAFRICA.

WHAT IS YOUR EDUCATIONAL BACKGROUND OR LEVEL? UNIVERSITY

PLEASE ENTER YOUR PRIMARY CONTACT NUMBER WITH COUNTRY CODE.GH+2330249649678

A Curriculum Vitae (C.V)
1) Personal Data: A Student of Ghana Christian University College.Advancing excellence in transforming leadership (Amrahia-Dodowa Rd.)
(Accredited by National Board of The Ministry Of Education)
Advancing Excellence in Transformational Leadership (Since 1966)

2)Surname:Andzie First name:Silas other name:Kwamena
3)Age:45 
4)Sex:Male
5)Qualification And Certificate of Appreciation
And Transcript:
a)A.G.P.A.:3.03  Enrole School Of Theology And Ministry 
Bachalor of Degree.Ghana Christian University College

b)Transfer to School Of Medicine Bachalor Of Degree.Metropolitan University College Of Medicine

c) Awarded Certificate Of Appreciation dedicated service as the Chaplain of Ghana Christian University College.
Students'Representative Council.(2006-2008)

d)Awarded Certificate Of Attendance &quot;Preparing Apostles for the Apostolic Season&quot;Annual Congress November 2006 by Abundant Life Ministries International.

e) Awards Certificate of Participation Holy Spirit Conference&quot;Restoration Of True Pastors by the Mystery Of the Body Of Christ Ministry.(October 2004)

f)Awarded Certificate Of Parcipation by Ghana Christian University College.&quot;Strategic Planning&quot; May 2007.

g)Awarded Certificate Of Participation &quot;Restoring True Spiritual Fatherhood&quot;Leadership Conference April 2006 by World Changers International.

h)Awarded Certificate of Bible Correspondence Course the Gospel Of St.John 96.84% Average Marks 48.42 by &quot;The Gospel Messagers&quot;Ministry.

5) Working
Experience:

i)Certificate of Incorporation
 Rescued Community Church             
   International issued for  service December 2014 and 

ii)Certificate To Commence
     Business 
Rescued Community Church International issued for service June 2015.

6)Chief Internal a)Auditor:Adjei-Nimako&amp;Associates
 b)Finance,Marketing Management Consultants
c) Secretary details:
Silas Kwamena Andzie

References:Rev.Silas Andzie,Founder &amp; President  of Rescued Community Church International and the support Ministries Department Current Account.Access Bank.

Mr.Godfred Yaw Oto.Korle-bu Heart Cardio Centre and Copy Media Printers Manager.

Pastor Augustine Kwaku Adjei.Associate Head Branch Redemption Hour Faith Ministries and Electronic Manager.

RESCUED COMMUNITY CHURCH INTERNATIONAL

Dear Sir|Madam,
LETTER OF RECOMMENDATION APPROVAL FOR LOCAL SPONSORSHIP TO DO STUDY ABROAD AND MINISTRY

The above stated company's name with the HeadPastor politely write to recommend approval for sponsorship acceptance letter to proceed study abroad.
  
However,through critical observation search made by the admission letters submitted to both schools.Namely,Ghana Christian University College and the Metropolitan University College Of Medicine and the curriculum vitae (CV)is witnessing and proofing the evidence account that deserve to be recommended to carryout this carrier goal in Medicine and Ministry.

We will therefore be glad to hear from the administrative body of Ghana National Petroleum Corporation to accept this offer with good will in order,to pay away to get local sponsorship to proceed study outside.</text>
      <sha1>6v6att1e4k2aqkbubfv1fwkm6mivbe3</sha1>
    </revision>
    <revision>
      <id>2624596</id>
      <parentid>2624593</parentid>
      <timestamp>2024-05-02T13:55:14Z</timestamp>
      <contributor>
        <username>Silas Andzie</username>
        <id>2970181</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4028" xml:space="preserve">Globe Education Centre-Travelling Abroad furthering studies.
Sunny FM-Direction to outfit
1Gempa Legon 
2Toyota Ghana UG opposite ask No.4
0244783000-0548393600
Return Month to call April back

GHANA EDUCATION SERVICE 

GETFUND 

OFFICIAL NUMBER:
0544323020

OFFICIAL WEBSITE
scholarship.get fund.gov.gh
------------------------------

 METROPOLITAN UNIVERSITY 
   COLLEGE OF MEDICINE

ADMISSION LETTER

SCHOLARSHIP $40%

CONTACT:+1(1813)-755-6705-
+1(268)562-9262-+(268)714-4680
info@muantigua.org
admissions@muantigua.org
Sir.Ryan's building,Newgate street st.John's Antigua,Antigua and Bartuda

FULLNAME:SILAS KWAMENA ANDZIE

EMAIL:Sandzie79@yahoo.|Silasandzie@gmail.com

PHONE NUMBER:GH+233 0249649678

DATE OF BIRTH :17-03-1979| 08-19-1987

WHICH COUNTRY ARE YOU FROM? GHANA-WESTAFRICA.

WHAT IS YOUR EDUCATIONAL BACKGROUND OR LEVEL? UNIVERSITY

PLEASE ENTER YOUR PRIMARY CONTACT NUMBER WITH COUNTRY CODE.GH+2330249649678

A Curriculum Vitae (C.V)
1) Personal Data: A Student of Ghana Christian University College.Advancing excellence in transforming leadership (Amrahia-Dodowa Rd.)
(Accredited by National Board of The Ministry Of Education)
Advancing Excellence in Transformational Leadership (Since 1966)

2)Surname:Andzie First name:Silas other name:Kwamena
3)Age:45 
4)Sex:Male
5)Qualification And Certificate of Appreciation
And Transcript:
a)A.G.P.A.:3.03  Enrole School Of Theology And Ministry 
Bachalor of Degree.Ghana Christian University College

b)Transfer to School Of Medicine Bachalor Of Degree.Metropolitan University College Of Medicine

c) Awarded Certificate Of Appreciation dedicated service as the Chaplain of Ghana Christian University College.
Students'Representative Council.(2006-2008)

d)Awarded Certificate Of Attendance &quot;Preparing Apostles for the Apostolic Season&quot;Annual Congress November 2006 by Abundant Life Ministries International.

e) Awards Certificate of Participation Holy Spirit Conference&quot;Restoration Of True Pastors by the Mystery Of the Body Of Christ Ministry.(October 2004)

f)Awarded Certificate Of Parcipation by Ghana Christian University College.&quot;Strategic Planning&quot; May 2007.

g)Awarded Certificate Of Participation &quot;Restoring True Spiritual Fatherhood&quot;Leadership Conference April 2006 by World Changers International.

h)Awarded Certificate of Bible Correspondence Course the Gospel Of St.John 96.84% Average Marks 48.42 by &quot;The Gospel Messagers&quot;Ministry.

5) Working
Experience:

i)Certificate of Incorporation
 Rescued Community Church             
   International issued for  service December 2014 and 

ii)Certificate To Commence
     Business 
Rescued Community Church International issued for service June 2015.

6)Chief Internal a)Auditor:Adjei-Nimako&amp;Associates
 b)Finance,Marketing Management Consultants
c) Secretary details:
Silas Kwamena Andzie

References:Rev.Silas Andzie,Founder &amp; President  of Rescued Community Church International and the support Ministries Department Current Account.Access Bank.

Mr.Godfred Yaw Oto.Korle-bu Heart Cardio Centre and Copy Media Printers Manager.

Pastor Augustine Kwaku Adjei.Associate Head Branch Redemption Hour Faith Ministries and Electronic Manager.

RESCUED COMMUNITY CHURCH INTERNATIONAL

Dear Sir|Madam,
LETTER OF RECOMMENDATION APPROVAL FOR LOCAL SPONSORSHIP TO DO STUDY ABROAD AND MINISTRY

The above stated company's name with the HeadPastor politely write to recommend approval for sponsorship acceptance letter to proceed study abroad.
  
However,through critical observation search made by the admission letters submitted to both schools.Namely,Ghana Christian University College and the Metropolitan University College Of Medicine and the curriculum vitae (CV)is witnessing and proven the evidence account that deserve to be recommended to carryout this carrier goal in Medicine and Ministry.

We will therefore be glad to hear from the administrative body of Ghana National Petroleum Corporation to accept this offer with good will in order,to pay away to get local sponsorship to proceed study outside.</text>
      <sha1>dflqshae6bshevg4z32i5qsij52e6yx</sha1>
    </revision>
    <revision>
      <id>2624708</id>
      <parentid>2624596</parentid>
      <timestamp>2024-05-02T15:19:54Z</timestamp>
      <contributor>
        <username>Silas Andzie</username>
        <id>2970181</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4085" xml:space="preserve">Globe Education Centre-Travelling Abroad furthering studies.
Sunny FM-Direction to outfit
1Gempa Legon 
2Toyota Ghana UG opposite ask No.4
0244783000-0548393600
Return Month to call April back

GHANA EDUCATION SERVICE 

GETFUND 

OFFICIAL NUMBER:
0544323020

OFFICIAL WEBSITE
scholarship.get fund.gov.gh
------------------------------

 METROPOLITAN UNIVERSITY 
   COLLEGE OF MEDICINE

ADMISSION LETTER

SCHOLARSHIP $40%

CONTACT:+1(1813)-755-6705-
+1(268)562-9262-+(268)714-4680
info@muantigua.org
admissions@muantigua.org
Sir.Ryan's building,Newgate street st.John's Antigua,Antigua and Bartuda

FULLNAME:SILAS KWAMENA ANDZIE

EMAIL:Sandzie79@yahoo.|Silasandzie@gmail.com

PHONE NUMBER:GH+233 0249649678

DATE OF BIRTH :17-03-1979| 08-19-1987

WHICH COUNTRY ARE YOU FROM? GHANA-WESTAFRICA.

WHAT IS YOUR EDUCATIONAL BACKGROUND OR LEVEL? UNIVERSITY

PLEASE ENTER YOUR PRIMARY CONTACT NUMBER WITH COUNTRY CODE.GH+2330249649678

A Curriculum Vitae (C.V)
1) Personal Data: A Student of Ghana Christian University College.Advancing excellence in transforming leadership (Amrahia-Dodowa Rd.)
(Accredited by National Board of The Ministry Of Education)
Advancing Excellence in Transformational Leadership (Since 1966)

2)Surname:Andzie First name:Silas other name:Kwamena
3)Age:45 
4)Sex:Male
5)Qualification And Certificate of Appreciation
And Transcript:
a)A.G.P.A.:3.03  Enrole School Of Theology And Ministry 
Bachalor of Degree.Ghana Christian University College

b)Transfer to School Of Medicine Bachalor Of Degree.Metropolitan University College Of Medicine

c) Awarded Certificate Of Appreciation dedicated service as the Chaplain of Ghana Christian University College.
Students'Representative Council.(2006-2008)

d)Awarded Certificate Of Attendance &quot;Preparing Apostles for the Apostolic Season&quot;Annual Congress November 2006 by Abundant Life Ministries International.

e) Awards Certificate of Participation Holy Spirit Conference&quot;Restoration Of True Pastors by the Mystery Of the Body Of Christ Ministry.(October 2004)

f)Awarded Certificate Of Parcipation by Ghana Christian University College.&quot;Strategic Planning&quot; May 2007.

g)Awarded Certificate Of Participation &quot;Restoring True Spiritual Fatherhood&quot;Leadership Conference April 2006 by World Changers International.

h)Awarded Certificate of Bible Correspondence Course the Gospel Of St.John 96.84% Average Marks 48.42 by &quot;The Gospel Messagers&quot;Ministry.

5) Working
Experience:

i)Certificate of Incorporation
 Rescued Community Church             
   International issued for  service December 2014 and 

ii)Certificate To Commence
     Business 
Rescued Community Church International issued for service June 2015.

6)Chief Internal a)Auditor:Adjei-Nimako&amp;Associates
 b)Finance,Marketing Management Consultants
c) Secretary details:
Silas Kwamena Andzie

References:Rev.Silas Andzie,Founder &amp; President  of Rescued Community Church International and the support Ministries Department Current Account.Access Bank.

Mr.Godfred Yaw Oto.Korle-bu Heart Cardio Centre and Copy Media Printers Manager.

Pastor Augustine Kwaku Adjei.Associate Head Branch Redemption Hour Faith Ministries and Electronic Manager.

RESCUED COMMUNITY CHURCH INTERNATIONAL

Dear Sir|Madam,
LETTER OF RECOMMENDATION APPROVAL FOR LOCAL SPONSORSHIP TO DO STUDY ABROAD AND MINISTRY

The above stated company's name with the HeadPastor politely write to recommend approval for sponsorship acceptance letter to proceed study abroad.
  
However,through critical observation search made by the admission letters submitted to both schools.Namely,Ghana Christian University College and the Metropolitan University College Of Medicine and the curriculum vitae (CV)is witnessing and proven the evidence account that deserve to be recommended to carryout this carrier goal in Medicine and Ministry.

We will therefore be glad to hear from the administrative body of Ghana National Petroleum Corporation to accept this offer with good will in order,to pay away to get local sponsorship to proceed study outside.

Yours Faithful

-------------

Silas Andzie
HeadPastor.</text>
      <sha1>fsfpob3uidefudfwqjm0spy0x2hhpla</sha1>
    </revision>
  </page>
  <page>
    <title>BIM-224 Research Infrastructures 24</title>
    <ns>0</ns>
    <id>304576</id>
    <revision>
      <id>2624582</id>
      <parentid>2624570</parentid>
      <timestamp>2024-05-02T12:11:17Z</timestamp>
      <contributor>
        <username>Lenskatala112358</username>
        <id>2981361</id>
      </contributor>
      <comment>/* Homework presentations: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4016" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Restaging-Fashion/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / presentation link (google slides, other slide platform, or wiki pages with screenshots)
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===


=== FINAL SUBMISSION ===</text>
      <sha1>ptemmy21uwlt4gu6jeb88vs3o5mzl4x</sha1>
    </revision>
    <revision>
      <id>2624709</id>
      <parentid>2624582</parentid>
      <timestamp>2024-05-02T15:27:07Z</timestamp>
      <contributor>
        <ip>2.205.33.182</ip>
      </contributor>
      <comment>/* Session 5: Data upload and querying (cont.) / Data visualisation and presentation */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4141" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Restaging-Fashion/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===

Slides are available here: https://docs.google.com/presentation/d/1wXePKwG7BxWUSjwL6WXBHYFCZrK6oT1106KM1-aHNU8/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== FINAL SUBMISSION ===</text>
      <sha1>cuzub4hhqdbb5kxj026689ogb8xpuue</sha1>
    </revision>
    <revision>
      <id>2624865</id>
      <parentid>2624709</parentid>
      <timestamp>2024-05-02T23:44:17Z</timestamp>
      <contributor>
        <username>Enesaak</username>
        <id>2981359</id>
      </contributor>
      <comment>/* Homework presentations: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4234" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Restaging-Fashion/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]
* Enes | [[:de:Kurs:OpenKnowledge24/EnesUnterlagen#/media/Datei:GoetheDiagramm.png|Diagram]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===

Slides are available here: https://docs.google.com/presentation/d/1wXePKwG7BxWUSjwL6WXBHYFCZrK6oT1106KM1-aHNU8/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== FINAL SUBMISSION ===</text>
      <sha1>7ymsmu2aaclajjsgurkb8wfhqeaime9</sha1>
    </revision>
    <revision>
      <id>2624957</id>
      <parentid>2624865</parentid>
      <timestamp>2024-05-03T09:41:13Z</timestamp>
      <contributor>
        <ip>2A02:560:4CE0:E500:D9E4:D35F:6E03:206</ip>
      </contributor>
      <comment>/* Group task 2 */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4246" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Historische-Portraetaufnahmen/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]
* Enes | [[:de:Kurs:OpenKnowledge24/EnesUnterlagen#/media/Datei:GoetheDiagramm.png|Diagram]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===

Slides are available here: https://docs.google.com/presentation/d/1wXePKwG7BxWUSjwL6WXBHYFCZrK6oT1106KM1-aHNU8/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== FINAL SUBMISSION ===</text>
      <sha1>k8h1ms062go8dgux3uouqzam8uame7i</sha1>
    </revision>
    <revision>
      <id>2624963</id>
      <parentid>2624957</parentid>
      <timestamp>2024-05-03T11:29:16Z</timestamp>
      <contributor>
        <username>Davud Kilic</username>
        <id>2983088</id>
      </contributor>
      <comment>/* Homework presentations: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4257" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Historische-Portraetaufnahmen/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]
* Davud I 

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]
* Enes | [[:de:Kurs:OpenKnowledge24/EnesUnterlagen#/media/Datei:GoetheDiagramm.png|Diagram]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===

Slides are available here: https://docs.google.com/presentation/d/1wXePKwG7BxWUSjwL6WXBHYFCZrK6oT1106KM1-aHNU8/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== FINAL SUBMISSION ===</text>
      <sha1>kbb0bn1y6d156wtmjnh40q9ilyl4e7a</sha1>
    </revision>
    <revision>
      <id>2624965</id>
      <parentid>2624963</parentid>
      <timestamp>2024-05-03T11:35:44Z</timestamp>
      <contributor>
        <username>Davud Kilic</username>
        <id>2983088</id>
      </contributor>
      <comment>/* Homework presentations: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4323" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Historische-Portraetaufnahmen/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Enes | [[:de:Datei:DMUSD_DataCleaning.pdf|Der Mensch und seine Dinge]]
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]
* Davud I [[:File:Presenation Davud Kilic.pdf|Schlangenköpfe und -körper]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Lena | [[:File:Update LMHoppe Schriftproben BGBM.pdf|Update Schriftprobensammlung BGBM]]
* Lena | [[:File:LMHoppe datamodel connections.png|data represenation in WikiData]]
* Enes | [[:de:Kurs:OpenKnowledge24/EnesUnterlagen#/media/Datei:GoetheDiagramm.png|Diagram]]

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===

Slides are available here: https://docs.google.com/presentation/d/1wXePKwG7BxWUSjwL6WXBHYFCZrK6oT1106KM1-aHNU8/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / wiki page
* ...

=== FINAL SUBMISSION ===</text>
      <sha1>bvbj6iltp37q80xminq52qyb4w5v6ri</sha1>
    </revision>
  </page>
  <page>
    <title>User:Addemf/sandbox/Technical Reasoning/Examples in Axiomatic Geometry</title>
    <ns>2</ns>
    <id>304948</id>
    <revision>
      <id>2624767</id>
      <parentid>2623855</parentid>
      <timestamp>2024-05-02T17:22:51Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="440" xml:space="preserve">Our previous study of sets served two purposes.  One was to know enough set theory to be able to use it in the study of other topics.  

The other was that it provided an opportunity for some relatively simple proofs.  These can serve as examples later on in the study of logic.

Before studying logic, it would be nice to see proofs in at least one other setting.  For the sake of variety, let's see some proofs in the setting of geometry.</text>
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      <text bytes="24016" xml:space="preserve">Our previous study of sets served two purposes.  One was to know enough set theory to be able to use it in the study of other topics.  

The other was that it provided an opportunity for some relatively simple proofs.  These can serve as examples later on in the study of logic.

Before studying logic, it would be nice to see proofs in at least one other setting.  For the sake of variety, let's see some proofs in the setting of geometry.

== Euclidean Geometry ==

Recall from the previous lesson that we saw a proof that the interior angles of a triangle sum to 180°.  After this proof, we saw that we will need to accept some statements as fundamental and will not require a proof of them.  Such a statement, we call an &quot;axiom&quot;.

We ended with the question &quot;Which statements will we accept as axioms?&quot;  Of course in some sense we must pick axioms which are designed to give us the geometry that we know we want.  

One of the most important results from Euclidean geometry is the Pythagorean theorem.  Let us see a proof of this theorem, and try to work backwards from this highly desirable result.  By thinking about what it is that we ''want'' from a theory, we are guided in trying to organize it and choose principles which deliver the thing we want.

Here is a beautiful and intuitive picture proof of the Pythagorean theorem.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot;
|
|-
| [[File:Pythagorean Theorem Proof, Construction Step 1.gif|left|200px|From any right triangle, extend ''a'' to a length ''b''.  Construct the perpendicular through the end, and extend a segment of length ''a''.  Complete the triangle, which is a copy of the original.]][[File:Pythagorean Theorem Proof, Further Construction Steps.gif|right|200px|Repeat the first construction step on the new triangle until arriving back to a vertex of the initial triangle.  Mark triangle and square regions.]]
|}

The way this works is:  

# Start from any right triangle, call it &lt;math&gt;\Delta ABC&lt;/math&gt; where ''C'' is the vertex at which the sides make a right angle.  Make four copies of the original triangle in the following way:  
## Extend &lt;math&gt;\overline{CB}&lt;/math&gt; by a length equal to ''AC''.  Call the end of this extension ''C'''.  
## Through &lt;math&gt;C'&lt;/math&gt; and perpendicular to &lt;math&gt;\overleftrightarrow{CB}&lt;/math&gt;, draw a segment of length equal to ''CB''.  Call the end of this segment &lt;math&gt;B'&lt;/math&gt;.  
## Triangle &lt;math&gt;\Delta BC'B'&lt;/math&gt; is now congruent to &lt;math&gt;\Delta ABC&lt;/math&gt;.  
## Repeat the above procedure on triangle &lt;math&gt;\Delta BC'B'&lt;/math&gt; to generate a new triangle.
## Repeat the procedure again on the new triangle.  
# This creates four triangles all congruent to the original triangle.  One can prove that the outer edges of these triangles now form a square ringing the entire figure.  
{{robelbox|title=Exercise 1. Square Area, Part I}}
Show that the square surrounding the figure constructed above has area 
: &lt;math&gt;(a+b)^2&lt;/math&gt;

Note that this course is designed to ''increase'' in rigor.  At this initial point in the course, proofs are meant to be compelling but not extremely rigorous or formal.  Just give a convincing explanation.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| Any given side of the square is composed of two disjoint segments, of lengths ''a'' and ''b'' respectively.  Therefore its area, by the area formula for rectangles, is &lt;math&gt;(a+b)^2&lt;/math&gt;.
|}
{{robelbox/close}}
{{robelbox|title=Exercise 2. Square Area, Part II}}
Notice that the figure also has an inner square, with sides formed from the hypotenuses of all of the copied triangles.  Argue that this square has area &lt;math&gt;c^2&lt;/math&gt;.

Moreover notice that, because the four triangles are congruent then they have the same area.  Also notice that the area of the entire figure is just the sum of the inner square and these four triangles.

Therefore argue that the area of the square is also equal to 
:&lt;math&gt;c^2+2ab&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| One can confirm that the inner square, really is a square, by checking that these sides meet at right angles.  To do so, one could use the knowledge that lines are perpendicular when their slopes are negative reciprocals of each other.  In this case, one can see that the hypotenuse of the original triangle has slope &lt;math&gt;-b/a&lt;/math&gt; while the hypotenuse of the first copy has slope &lt;math&gt;a/b&lt;/math&gt;.  

Therefore all edges of the inner figure meet at right angles and have the same length ''c'', hence it is a square.  Therefore it has area &lt;math&gt;c^2&lt;/math&gt;.
|}
{{robelbox/close}}
{{robelbox|title=Exercise 3. Pythagorean Theorem}}
Use the two exercises above to prove the Pythagorean theorem
: &lt;math&gt;a^2+b^2=c^2&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| From the two exercises above, the same area is both &lt;math&gt;(a+b)^2&lt;/math&gt; and &lt;math&gt;c^2+2ab&lt;/math&gt;, therefore 
: &lt;math&gt;(a+b)^2=c^2+2ab&lt;/math&gt;

But by distribution, &lt;math&gt;(a+b)^2 = a^2 + 2ab + b^2&lt;/math&gt; and therefore 
: &lt;math&gt;a^2+2ab+b^2 = c^2+2ab&lt;/math&gt;

Then by subtracting &lt;math&gt;2ab&lt;/math&gt; we have the Pythagorean theorem.
: &lt;math&gt;a^2+b^2 = c^2&lt;/math&gt;
|}
{{robelbox/close}}

Certainly the proof above assumes the possibility of extending a line segment by a given length.  That seems like a good candidate for an axiom, since it sounds quite fundamental and probably cannot be proved from any simpler statements.  

Here are other assumptions made by the proof.

* Given a point ''P'' and line &lt;math&gt;\ell&lt;/math&gt;, there is always a line through ''P'' and perpendicular to &lt;math&gt;\ell&lt;/math&gt;.
* Through any two points is a line segment connecting them.
* If the segments &lt;math&gt;\overline{WX},\overline{XY}, \overline{YZ},\overline{YZ'}&lt;/math&gt; are all of the same length, at right angles to each other, and have the same &quot;orientation&quot; (i.e. every angle formed is, so-to-speak, a &quot;left turn&quot;), then &lt;math&gt;Z=Z'&lt;/math&gt;.  
* The length of segments on a line are additive:  If &lt;math&gt;\overline{ABC}&lt;/math&gt; is a segment, then its length, ''AC'' is the same as the sum of the component lengths, &lt;math&gt;AB+BC&lt;/math&gt;.  
* The area formulas for triangles.
* Area is additive, in a way analogous to how length is additive.  

Not all of these seem like equally good candidates for being axioms.  They all seem relatively compelling, I would argue -- but some of them, like the third bullet point above, seems much too complex to be an axiom.  Axioms should be relatively simple, fundamental, and not provable from other simpler propositions.  That third bullet-point certainly ''feels'' as though it should be provable if we chose the right collection of other, simpler axioms.  

The ancient Greeks were aware of proofs like this one and others.  In an effort to bring all of these various facts into a single, unified, and organized body, Euclid proposed a collection of axioms from which all of geometry could be proved.   

Euclid wrote down only a few axioms, which were

# To draw a straight line from any point to any point.
# To produce (extend) a finite straight line continuously in a straight line.
# To describe a circle with any centre and distance (radius).
# That all right angles are equal to one another.
# [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. 

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot;
| 
|-
| [[File:Parallel Postulate Animation.gif|left|400px|Animation of the objects referred to in Euclid's parallel postulate.]]
|}

This last one is quite complicated, so above is an animation demonstrating its meaning.  First two lines are shown and then two points, one on each.  The segment between them is constructed, and then the angles on one side of the segment.  Because these angles sum to less than 180° then the lines intersect on this side.

More information about Euclid's work can be found at the Wikipedia page [https://en.wikipedia.org/wiki/Euclidean_geometry| Euclidean geometry].

=== Hilbert-style Axioms ===

Ultimately it turns out that Euclid's axioms are not quite adequate for the level of rigor and completeness that we would like.  In more recent times, a more precise and complete set of axioms were laid out by various mathematicians.  

One of the more famous of these was [https://en.wikipedia.org/wiki/Hilbert's_axioms| Hilbert's axioms].  This consisted of 20 axioms, and in order not to get too distracted, we won't try to discuss all of them or get too deep into the study of geometry.  

However, it will be instructive to at least prove a few of the most basic results of geometry by means of Hilbert-styled axioms.

As a general note about language, we will talk about points being &quot;on&quot; or &quot;in&quot; a line.  Equivalently we may say that that the line &quot;runs through&quot; a point, or &quot;contains&quot; a point.  This will be defined after we have precisely defined the word &quot;line&quot;.

Here is a first axiom:
{{robelbox|title=Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Notice that this axiom refers to objects which we have not defined.  

As we have said repeatedly, statements cannot have infinitely many proofs, and we must choose some foundational statements for axioms.  

Well the same, too, must be true of definitions as well!  While we can define some concepts in terms of other concepts, eventually we must have some collection of undefined objects.

Our first undefined concept is that of a '''point'''.  We will talk about sets of points, we will prove things about them, but we will not try to state what they are.  We take the idea of a point to be foundational and not in need of further definition.

In fact we will define what it means to be a line, in terms of points.  In fact we will define the concept of a line in terms of distances between points, but in order to that, we first have to define the concept of distance.  This will come shortly.  

We next define the '''plane''' as the set of all points, and we write this as &lt;math&gt;\mathfrak P&lt;/math&gt;.  
{{definition|name=plane|value=
The plane is the set &lt;math&gt;\mathfrak P&lt;/math&gt; of all points.  In set notation, 
: &lt;math&gt;\mathfrak P = \{q: q \text{ is a point}\}&lt;/math&gt;
}}

We will now define '''distance''' to be a relationship between points and a nonnegative real number.  We denote the distance between points ''A'' and ''B'' by the expression &lt;math&gt;d(A,B)&lt;/math&gt;.  

Note that its definition, below, is only that it is some unique nonnegative real number.  Which number it is for any given pair of points is left open-ended.  It will be further specified by axioms later.
[[File:Distance between.svg|thumb|A diagram labeling distances between points, some collinear and some not.  Collinear points have distances which are &quot;additive&quot; and non-collinear points do not.]]

{{definition|name=distance, ''d''|value=
The '''distance from ''A'' to ''B''''' is denoted &lt;math&gt;d(A,B)&lt;/math&gt; and is defined to be some unique nonnegative real number.
}}

In the following axiom we show that the distance must also satisfy certain properties that we would recognize, from anything worthy of the name &quot;distance&quot;.
{{robelbox|title=Distance Axiom|theme=2}}
For any two points, &lt;math&gt;A,B\in \mathfrak P&lt;/math&gt;, the distance from any point to itself is zero.
: &lt;math&gt;d(A,A)=0&lt;/math&gt;
Also, distance is commutative, which is to say, the distance from ''A'' to ''B'' is the same as the distance from ''B'' to ''A''.
: &lt;math&gt;d(A,B)=d(B,A)&lt;/math&gt;
{{robelbox/close}}

{{definition|name=between|value=
For distinct points &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt; we say that ''B'' is '''between''' ''A'' and ''B'', and we write &lt;math&gt;[ABC]&lt;/math&gt;, when 
: &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt;
Moreover, for any number of points, &lt;math&gt;3\le n&lt;/math&gt;, and the points themselves, &lt;math&gt;A_1,A_2,\dots,A_n\in \mathfrak P&lt;/math&gt;, we may write &lt;math&gt;[A_1A_2\dots A_n]&lt;/math&gt;.  Formally this is understood as shorthand for 
: &lt;math&gt;[A_1A_2A_3]&lt;/math&gt; and &lt;math&gt;[A_2A_3A_4]&lt;/math&gt; and ... and &lt;math&gt;[A_{n-2}A_{n-1}A_n]&lt;/math&gt;.  
}}
{{robelbox|title=Exercise 4. The commutativity of betweenness}}
Prove that if ''B'' is between ''A'' and ''C'' then it also follows that ''B'' is between ''C'' and ''A''.  

Note that, at this point, proofs are meant to be increasing in their rigor and therefore, in this proof, we do not want to rely on things which &quot;should intuitively be true&quot;.  Rather, we want the proof to come very formally and precisely from the exact statements of definitions and axioms.  

However, also note that we will be assuming that ''numbers'' work in the way that we are familiar with.  For example we will assume that &lt;math&gt;1+2=2+1&lt;/math&gt; without need for justification.  Later in the course we will actually investigate axiom systems for numbers, but for now we assume their familiar properties.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution 
|-
| Assume that ''B'' is between ''A'' and ''C''.  

By definition of &lt;math&gt;[ABC]&lt;/math&gt; we have 
: &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt;

By the ''Distance Axiom'', &lt;math&gt;d(A,C)=d(C,A)&lt;/math&gt; and &lt;math&gt;d(B,A)=d(A,B)&lt;/math&gt; and &lt;math&gt;d(B,C)=d(C,B)&lt;/math&gt;.  Therefore by substitution of numbers in an equation (something we  assume, as properties of numbers),
: &lt;math&gt;d(C,A) = d(B,A)+d(C,B)&lt;/math&gt;
Moreover, the sum of any two numbers is always commutative, meaning that we can swap their placement around the sum.
: &lt;math&gt;d(C,A) = d(C,B)+d(B,A)&lt;/math&gt;

But this is exactly the definition of &lt;math&gt;[CBA]&lt;/math&gt;.

&lt;math&gt;\Box&lt;/math&gt;

(Note that the &quot;box&quot; symbol, lovingly called the &quot;Halmos&quot; in honor of [https://en.wikipedia.org/wiki/Paul_Halmos| Paul Halmos], indicates that the desired conclusion has been reached, and therefore the proof has ended.) [[File:Paul Halmos 1986.jpg|thumb|Paul Halmos, ♥]]
|}
{{robelbox/close}}
{{definition|name=line|value=
For any two distinct points, &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, define the '''line through ''A'' and ''B''''' as the set of all points satisfying any &quot;betweenness&quot; relationship with ''A'' and ''B''.  We write &lt;math&gt;\overleftrightarrow{AB}&lt;/math&gt; to denote the line, and formally define it as the set 
: &lt;math&gt;\overleftrightarrow{AB}=\{A,B\}\cup \{C\in \mathfrak P: [CAB] \text{ or } [ACB]\text{ or } [ABC]\}&lt;/math&gt;

If any two points are on the same line, we say the two points are '''collinear'''.  
}}

Read this as saying &quot;The line through ''A'' and ''B'' is defined to be the set of all points, ''C'', such that either ''A'' is between ''C'' and ''B'', or ''C'' is between ''A'' and ''B'', or ''B'' is between ''A'' and ''C''.&quot;

At this point we have finally defined all of the objects referred to in the ''Two Point Axiom''!  With all of this ground cleared, we are actually in a position to state several more axioms, and the prove another lemma.
{{robelbox|title=Three Point Axiom|theme=2}}
There exist some three distinct points which are not all collinear.
{{robelbox/close}}
{{robelbox|title=Unique Between Axiom|theme=2}}
For any three distinct collinear points, &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt;, precisely one is between the other two.
{{robelbox/close}}
{{robelbox|title= Exercise . Unique up to reordering}}
Prove that for any three distinct colinear points &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt;, precisely one of the following holds,

* &lt;math&gt;[ABC]&lt;/math&gt;
* &lt;math&gt;[ACB]&lt;/math&gt;
* &lt;math&gt;[BAC]&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| From the ''Unique Between Axiom'' we know that precisely one of the three points is between the other two.  Let's assume ''B'' is between ''A'' and ''C'', which is to say, &lt;math&gt;[ABC]&lt;/math&gt;.

The the first bullet point holds, which already shows that at least one of the three possibilities holds.  

If &lt;math&gt;[ACB]&lt;/math&gt; were also true then ''C'' would be between ''A'' and ''B''.  But this would contract the uniqueness of the ''Unique Between Axiom'', so it is impossible.  

Likewise &lt;math&gt;[BAC]&lt;/math&gt; is impossible in the same way that we showed &lt;math&gt;[ACB]&lt;/math&gt; is impossible.  

This has now shown that if &lt;math&gt;[ABC]&lt;/math&gt; holds then the other two cannot.  ''Mutatis mutandis'', the same proof shows that if any one of the other two were true then it would prohibit the possibility of its others.  

□
|}
{{robelbox/close}}
{{robelbox|title=Infinity Axiom|theme=2}}
If &lt;math&gt;B,D\in\mathfrak P&lt;/math&gt; then there are points &lt;math&gt;A,C,E\in\mathfrak P&lt;/math&gt; such that &lt;math&gt;[ABCDE]&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise 5. Infinitely many points}}
Prove, using any of the axioms and definitions above, that there are infinitely many points.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| By the ''Three Points Axiom'' we know that there are some three points.  For now we will only need two of them, so call two of them &lt;math&gt;A_1&lt;/math&gt; and &lt;math&gt;A_2&lt;/math&gt;.

By the ''Two Points Axiom'' there is a line through these two points, &lt;math&gt;\overleftrightarrow{A_1A_2}&lt;/math&gt;.  

By the ''Infinity Axiom'' there must exist a point &lt;math&gt;A_3\in\mathfrak P&lt;/math&gt;, such that &lt;math&gt;[A_1A_2A_3]&lt;/math&gt;.  By definition of betweenness, &lt;math&gt;A_3&lt;/math&gt; must be distinct from each of &lt;math&gt;A_1,A_2&lt;/math&gt;.  So now there are three points.

Continuing likewise there must be a fourth point satisfying &lt;math&gt;[A_2A_3A_4]&lt;/math&gt;.  Continuing in the same manner, for any ''n'', there must be points &lt;math&gt;[A_1A_2\dots A_n]&lt;/math&gt;.  

Therefore, for any number ''M'', the number of points is not bounded by ''M''.  This is because there is a sequence of points &lt;math&gt;[A_1A_2\dots A_{M+1}]&lt;/math&gt;, all distinct from each other.  Then there are at least &lt;math&gt;M+1&lt;/math&gt; points and, as stated, therefore ''M'' does not bound the number of points.

Since the number of points is not bounded by any finite number, therefore it is infinite.

□
|}
{{robelbox/close}}
{{robelbox|title=Exercise 5. Distance definiteness}}
Use the axioms and definitions above to prove that, for any &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, we have 
: &lt;math&gt;d(A,B)=0&lt;/math&gt; if and only if &lt;math&gt;A=B&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| Suppose &lt;math&gt;A=B&lt;/math&gt;.  Then it follows immediately that &lt;math&gt;d(A,B) = d(A,A)&lt;/math&gt; and then by the ''Distance Axiom'' it follows that this is zero.

We now need to prove that, if &lt;math&gt;d(A,B)=0&lt;/math&gt; then therefore &lt;math&gt;A=B&lt;/math&gt;, and this will be the harder direction.  

If &lt;math&gt;d(A,B)=0&lt;/math&gt; then suppose that ''A'' is ''not B''.  Well in that case, these points are distinct, and the ''Two Point Axiom'' entails that there is a unique line through them, &lt;math&gt;\overleftrightarrow{AB}&lt;/math&gt;.  

Then by the ''Infinity Axiom'' there is a point between them, say &lt;math&gt;[ACB]&lt;/math&gt;.  Therefore these are three distinct collinear points.

As such, the ''Distance Axiom'' now tells us that 
: &lt;math&gt;d(A,B)=d(A,C)+d(C,B)&lt;/math&gt;

But since these are each nonnegative quantities, and &lt;math&gt;d(A,B)=0&lt;/math&gt; by assumption, then therefore &lt;math&gt;d(A,C)=0&lt;/math&gt; and &lt;math&gt;d(C,B)=0&lt;/math&gt;.

But now it follows that &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt; using the ''Distance Axiom'' to commute some of the points in these expressions.  

But this now entails, by definition of betweenness, that &lt;math&gt;[ABC]&lt;/math&gt;.

But this shows that, of the three distinct points ''A'', ''B'', and ''C'', both ''B'' is between its other points, and also ''C'' is between its other points.

But this contradicts the ''Unique Between Axiom''.

What this shows is that, when &lt;math&gt;d(A,B)=0&lt;/math&gt;, if we further assume that &lt;math&gt;A\ne B&lt;/math&gt; then this leads to a contradiction.  Therefore we must have that &lt;math&gt;A=B&lt;/math&gt;, just as we needed to prove.  

□
|}
{{robelbox/close}}
{{robelbox|title=Exercise 6. Rays}}
Now you do it!

With inspiration taken from how we defined a line, write a reasonable definition of a ''ray''.  

Recall that, for any two distinct points &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, the ray from ''A'' through ''B'' is supposed to be effectively half of the line through ''A'' and ''B''.  In particular, it is suppose to be the half of the line which contains both of these points, and is &quot;cut&quot; at ''A''.  

This is a half-formal definition of a ray. Your challenge is to re-state this, fully formally.

Likewise, define the ''segment'' from ''A'' to ''B''.

We will write the ray from ''A'' through ''B'' by &lt;math&gt;\overrightarrow{AB}&lt;/math&gt;.  We will write the segment from ''A'' to ''B'' with &lt;math&gt;\overline{AB}&lt;/math&gt;.

Once you have defined these formally, then prove the following.  For any two distinct points &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, 

: &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA} = \overline{AB}&lt;/math&gt;
and
: &lt;math&gt;\overrightarrow{AB}\cup\overrightarrow{BA} = \overleftrightarrow{AB}&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| The definition of &lt;math&gt;\overrightarrow{AB}&lt;/math&gt; is the set 
: &lt;math&gt;\overrightarrow{AB}=\{C\in\mathfrak P: [ACB] \text{ or } [ABC]\}&lt;/math&gt; 
and the definition of &lt;math&gt;\overline{AB}&lt;/math&gt; is 
: &lt;math&gt;\overline{AB}=\{C\in\mathfrak P: [ACB]\}&lt;/math&gt;

Now we prove that &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA} = \overline{AB}&lt;/math&gt;.

Notice that the thing we are to prove here, is an equation ''of sets''.  Therefore, as is standard when proving the equality of two sets, we start by proofing one subset direction.

Let us start with proving &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA}\subseteq\overline{AB}&lt;/math&gt;.  In order to do so, we assume &lt;math&gt;p\in \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt; and then attempt to show &lt;math&gt;p\in\overline{AB}&lt;/math&gt;.

From &lt;math&gt; p\in\overrightarrow{AB}&lt;/math&gt; we have that either &lt;math&gt;[ApB]&lt;/math&gt; or &lt;math&gt;[ABp]&lt;/math&gt;.  From &lt;math&gt;p\in\overrightarrow{BA}&lt;/math&gt; we have that either &lt;math&gt;[BpA]&lt;/math&gt; or &lt;math&gt;[BAp]&lt;/math&gt;.

If &lt;math&gt;[ABp]&lt;/math&gt; then it would follow that we cannot have &lt;math&gt;[BpA]&lt;/math&gt; because this is equivalent (as we proved in ''The commutativity of betweenness'') to &lt;math&gt;[ApB]&lt;/math&gt;.  This, in turn, is impossible because it would contradict the ''Uniqueness Axiom''.  In turn, that is because we would have both ''B'' and ''p'' as the points which are between the others.  

By a very similar argument, ''mutatis mutandis'', if &lt;math&gt;[ABp]&lt;/math&gt; then it would follow that we cannot have &lt;math&gt;[BAp]&lt;/math&gt;.

Therefore if &lt;math&gt;[ABp]&lt;/math&gt; holds then neither &lt;math&gt;[BpA]&lt;/math&gt; nor &lt;math&gt;[BAp]&lt;/math&gt;.  But this contradicts the earlier result that we must have one or the other.  Therefore, we cannot have &lt;math&gt;[ABp]&lt;/math&gt;.

Since we established earlier that either &lt;math&gt;[ApB]&lt;/math&gt; or &lt;math&gt;[ABp]&lt;/math&gt;, but we have just seen that we cannot have &lt;math&gt;[ABp]&lt;/math&gt;, then therefore we must have &lt;math&gt;[ApB]&lt;/math&gt;.  

But then immediate from this and the definition of the segment, we must have &lt;math&gt;p\in\overline{AB}&lt;/math&gt;.

We have now shown that &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA}\subseteq\overline{AB}&lt;/math&gt;.  We move on to show that &lt;math&gt;\overline{AB}\subseteq\overrightarrow{AB}\cap \overrightarrow{BA}&lt;/math&gt;.

Let &lt;math&gt;p\in\overline{AB}&lt;/math&gt;, so by definition, &lt;math&gt;[ApB]&lt;/math&gt;.

Then by definition &lt;math&gt;p\in\overrightarrow{AB}&lt;/math&gt;.  By the same definition, and from the result ''The commutativity of betweenness'' exercise above, then also, &lt;math&gt;p\in\overrightarrow{BA}&lt;/math&gt;.

Therefore &lt;math&gt;p\in\overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt; as desired, and so &lt;math&gt;\overline{AB}\subseteq \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt;

This concludes the proof that &lt;math&gt;\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt;.  

The proof that &lt;math&gt;\overrightarrow{AB}\cup\overrightarrow{BA}=\overleftrightarrow{AB}&lt;/math&gt; is very similar.

□
|}
{{robelbox/close}}</text>
      <sha1>pxzee2ka69sf7itzqn674fmbob2snrv</sha1>
    </revision>
    <revision>
      <id>2624772</id>
      <parentid>2624768</parentid>
      <timestamp>2024-05-02T18:12:00Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23892" xml:space="preserve">Our previous study of sets served two purposes.  One was to know enough set theory to be able to use it in the study of other topics.  

The other was that it provided an opportunity for some relatively simple proofs.  These can serve as examples later on in the study of logic.

Before studying logic, it would be nice to see proofs in at least one other setting.  For the sake of variety, let's see some proofs in the setting of geometry.

== Euclidean Geometry ==

=== The Search for Axioms ===

Recall from a previous lesson that we saw a proof that the interior angles of a triangle sum to 180°.  

After this proof, we reflected on the fact that we must accept some statements as fundamental.  These statements, which we accept without proof, are called &quot;axioms&quot;.

We ended with the question &quot;Which statements will we accept as axioms?&quot;  Of course in some sense we must pick axioms which are designed to give us the geometry that we know we want.  

One of the most important results from Euclidean geometry is the Pythagorean theorem.  Let us see a proof of this theorem, and try to work backwards from this highly desirable result.  

By thinking about what it is that we ''want'' from a theory, we are guided in trying to organize it and choose principles which deliver the thing we want.

=== The Pythagorean Theorem ===

Here is a beautiful and intuitive picture proof of the Pythagorean theorem.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Pythagorean Theorem Proof, Construction Step 1.gif|left|200px|From any right triangle, extend ''a'' to a length ''b''.  Construct the perpendicular through the end, and extend a segment of length ''a''.  Complete the triangle, which is a copy of the original.]]
|}

The way this works is:  

1. Start from any right triangle, call it &lt;math&gt;\Delta ABC&lt;/math&gt; where ''C'' is the vertex at which the sides make a right angle.  Make four copies of the original triangle in the following way:  
: 1. Extend &lt;math&gt;\overline{CB}&lt;/math&gt; by a length equal to ''AC''.  Call the end of this extension ''C'''.  
: 2. Through &lt;math&gt;C'&lt;/math&gt; and perpendicular to &lt;math&gt;\overleftrightarrow{CB}&lt;/math&gt;, draw a segment of length equal to ''CB''.  Call the end of this segment &lt;math&gt;B'&lt;/math&gt;.  
: 3. Triangle &lt;math&gt;\Delta BC'B'&lt;/math&gt; is now congruent to &lt;math&gt;\Delta ABC&lt;/math&gt;.  
: 4. Repeat the above procedure on triangle &lt;math&gt;\Delta BC'B'&lt;/math&gt; to generate a new triangle.
{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|-
| [[File:Pythagorean Theorem Proof, Further Construction Steps.gif|right|200px|Repeat the first construction step on the new triangle until arriving back to a vertex of the initial triangle.  Mark triangle and square regions.]]
|}
: 5. Repeat the procedure again on the new triangle.  

2. This creates four triangles all congruent to the original triangle.  One can prove that the outer edges of these triangles now form a square ringing the entire figure.  
{{robelbox|title=Exercise 1. Square Area, Part I}}
Show that the square surrounding the figure constructed above has area 
: &lt;math&gt;(a+b)^2&lt;/math&gt;

Note that this course is designed to ''increase'' in rigor.  At this initial point in the course, proofs are meant to be compelling but not extremely rigorous or formal.  Just give a convincing explanation.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| Any given side of the square is composed of two disjoint segments, of lengths ''a'' and ''b'' respectively.  Therefore its area, by the area formula for rectangles, is &lt;math&gt;(a+b)^2&lt;/math&gt;.
|}
{{robelbox/close}}
{{robelbox|title=Exercise 2. Square Area, Part II}}
Notice that the figure also has an inner square, with sides formed from the hypotenuses of all of the copied triangles.  Argue that this square has area &lt;math&gt;c^2&lt;/math&gt;.

Moreover notice that, because the four triangles are congruent then they have the same area.  Also notice that the area of the entire figure is just the sum of the inner square and these four triangles.

Therefore argue that the area of the square is also equal to 
:&lt;math&gt;c^2+2ab&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| One can confirm that the inner square, really is a square, by checking that these sides meet at right angles.  To do so, one could use the knowledge that lines are perpendicular when their slopes are negative reciprocals of each other.  In this case, one can see that the hypotenuse of the original triangle has slope &lt;math&gt;-b/a&lt;/math&gt; while the hypotenuse of the first copy has slope &lt;math&gt;a/b&lt;/math&gt;.  

Therefore all edges of the inner figure meet at right angles and have the same length ''c'', hence it is a square.  Therefore it has area &lt;math&gt;c^2&lt;/math&gt;.
|}
{{robelbox/close}}
{{robelbox|title=Exercise 3. Pythagorean Theorem}}
Use the two exercises above to prove the Pythagorean theorem
: &lt;math&gt;a^2+b^2=c^2&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| From the two exercises above, the same area is both &lt;math&gt;(a+b)^2&lt;/math&gt; and &lt;math&gt;c^2+2ab&lt;/math&gt;, therefore 
: &lt;math&gt;(a+b)^2=c^2+2ab&lt;/math&gt;

But by distribution, &lt;math&gt;(a+b)^2 = a^2 + 2ab + b^2&lt;/math&gt; and therefore 
: &lt;math&gt;a^2+2ab+b^2 = c^2+2ab&lt;/math&gt;

Then by subtracting &lt;math&gt;2ab&lt;/math&gt; we have the Pythagorean theorem.
: &lt;math&gt;a^2+b^2 = c^2&lt;/math&gt;
|}
{{robelbox/close}}

Certainly the proof above assumes the possibility of extending a line segment by a given length.  That seems like a good candidate for an axiom, since it sounds quite fundamental and probably cannot be proved from any simpler statements.  

Here are other assumptions made by the proof.

* Given a point ''P'' and line &lt;math&gt;\ell&lt;/math&gt;, there is always a line through ''P'' and perpendicular to &lt;math&gt;\ell&lt;/math&gt;.
* Through any two points is a line segment connecting them.
* If the segments &lt;math&gt;\overline{WX},\overline{XY}, \overline{YZ},\overline{YZ'}&lt;/math&gt; are all of the same length, at right angles to each other, and have the same &quot;orientation&quot; (i.e. every angle formed is, so-to-speak, a &quot;left turn&quot;), then &lt;math&gt;Z=Z'&lt;/math&gt;.  
* The length of segments on a line are additive:  If &lt;math&gt;\overline{ABC}&lt;/math&gt; is a segment, then its length, ''AC'' is the same as the sum of the component lengths, &lt;math&gt;AB+BC&lt;/math&gt;.  
* The area formulas for triangles.
* Area is additive, in a way analogous to how length is additive.  

Not all of these seem like equally good candidates for being axioms.  They all seem relatively compelling, I would argue -- but some of them, like the third bullet point above, seems much too complex to be an axiom.  Axioms should be relatively simple, fundamental, and not provable from other simpler propositions.  That third bullet-point certainly ''feels'' as though it should be provable if we chose the right collection of other, simpler axioms.  

The ancient Greeks were aware of proofs like this one and others.  In an effort to bring all of these various facts into a single, unified, and organized body, Euclid proposed a collection of axioms from which all of geometry could be proved.   

Euclid wrote down only a few axioms, which were

# To draw a straight line from any point to any point.
# To produce (extend) a finite straight line continuously in a straight line.
# To describe a circle with any centre and distance (radius).
# That all right angles are equal to one another.
# [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. 

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot;
| 
|-
| [[File:Parallel Postulate Animation.gif|left|400px|Animation of the objects referred to in Euclid's parallel postulate.]]
|}

This last one is quite complicated, so above is an animation demonstrating its meaning.  First two lines are shown and then two points, one on each.  The segment between them is constructed, and then the angles on one side of the segment.  Because these angles sum to less than 180° then the lines intersect on this side.

More information about Euclid's work can be found at the Wikipedia page [https://en.wikipedia.org/wiki/Euclidean_geometry| Euclidean geometry].

=== Hilbert-style Axioms ===

Ultimately it turns out that Euclid's axioms are not quite adequate for the level of rigor and completeness that we would like.  In more recent times, a more precise and complete set of axioms were laid out by various mathematicians.  

One of the more famous of these was [https://en.wikipedia.org/wiki/Hilbert's_axioms| Hilbert's axioms].  This consisted of 20 axioms, and in order not to get too distracted, we won't try to discuss all of them or get too deep into the study of geometry.  

However, it will be instructive to at least prove a few of the most basic results of geometry by means of Hilbert-styled axioms.

As a general note about language, we will talk about points being &quot;on&quot; or &quot;in&quot; a line.  Equivalently we may say that that the line &quot;runs through&quot; a point, or &quot;contains&quot; a point.  This will be defined after we have precisely defined the word &quot;line&quot;.

Here is a first axiom:
{{robelbox|title=Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Notice that this axiom refers to objects which we have not defined.  

As we have said repeatedly, statements cannot have infinitely many proofs, and we must choose some foundational statements for axioms.  

Well the same, too, must be true of definitions as well!  While we can define some concepts in terms of other concepts, eventually we must have some concepts which are not defined.

Our first undefined concept is that of a '''point'''.  We will talk about sets of points, we will prove things about them, but we will not try to state what they are.  We take the idea of a point to be foundational and not in need of further definition.

In fact, we will define lines in terms of distances between points.  

In order to that, we first have to define the concept of distance.  This will come shortly.  

We next define the '''plane''' as the set of all points, and we write this as &lt;math&gt;\mathfrak P&lt;/math&gt;.  
{{definition|name=plane|value=
The plane is the set &lt;math&gt;\mathfrak P&lt;/math&gt; of all points.  In set notation, 
: &lt;math&gt;\mathfrak P = \{q: q \text{ is a point}\}&lt;/math&gt;
}}

We will now define '''distance''' to be a relationship between points and a nonnegative real number.  We denote the distance between points ''A'' and ''B'' by the expression &lt;math&gt;d(A,B)&lt;/math&gt;.  
[[File:Distance between.svg|thumb|A diagram labeling distances between points, some collinear and some not.  Collinear points have distances which are &quot;additive&quot; and non-collinear points do not.]]

{{definition|name=distance, ''d''|value=
The '''distance from ''A'' to ''B''''' is denoted &lt;math&gt;d(A,B)&lt;/math&gt; and is defined to be some unique nonnegative real number.
}}

In the following axiom we show that the distance must also satisfy certain properties that we would recognize, from anything worthy of the name &quot;distance&quot;.
{{robelbox|title=Distance Axiom|theme=2}}
For any two points, &lt;math&gt;A,B\in \mathfrak P&lt;/math&gt;, the distance from any point to itself is zero.
: &lt;math&gt;d(A,A)=0&lt;/math&gt;
Also, distance is commutative, which is to say, the distance from ''A'' to ''B'' is the same as the distance from ''B'' to ''A''.
: &lt;math&gt;d(A,B)=d(B,A)&lt;/math&gt;
{{robelbox/close}}

{{definition|name=between|value=
For distinct points &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt; we say that ''B'' is '''between''' ''A'' and ''B'', and we write &lt;math&gt;[ABC]&lt;/math&gt;, when 
: &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt;
Moreover, for any number of points, &lt;math&gt;3\le n&lt;/math&gt;, and the points themselves, &lt;math&gt;A_1,A_2,\dots,A_n\in \mathfrak P&lt;/math&gt;, we may write &lt;math&gt;[A_1A_2\dots A_n]&lt;/math&gt;.  Formally this is understood as shorthand for 
: &lt;math&gt;[A_1A_2A_3]&lt;/math&gt; and &lt;math&gt;[A_2A_3A_4]&lt;/math&gt; and ... and &lt;math&gt;[A_{n-2}A_{n-1}A_n]&lt;/math&gt;.  
}}
{{robelbox|title=Exercise 4. The commutativity of betweenness}}
Prove that if ''B'' is between ''A'' and ''C'' then it also follows that ''B'' is between ''C'' and ''A''.  

Note that, at this point, proofs are meant to be increasing in their rigor and therefore, in this proof, we do not want to rely on things which &quot;should intuitively be true&quot;.  Rather, we want the proof to come very formally and precisely from the exact statements of definitions and axioms.  

However, also note that we will be assuming that ''numbers'' work in the way that we are familiar with.  For example we will assume that &lt;math&gt;1+2=2+1&lt;/math&gt; without need for justification.  Later in the course we will actually investigate axiom systems for numbers, but for now we assume their familiar properties.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution 
|-
| Assume that ''B'' is between ''A'' and ''C''.  

By definition of &lt;math&gt;[ABC]&lt;/math&gt; we have 
: &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt;

By the ''Distance Axiom'', &lt;math&gt;d(A,C)=d(C,A)&lt;/math&gt; and &lt;math&gt;d(B,A)=d(A,B)&lt;/math&gt; and &lt;math&gt;d(B,C)=d(C,B)&lt;/math&gt;.  Therefore by substitution of numbers in an equation (something we  assume, as properties of numbers),
: &lt;math&gt;d(C,A) = d(B,A)+d(C,B)&lt;/math&gt;
Moreover, the sum of any two numbers is always commutative, meaning that we can swap their placement around the sum.
: &lt;math&gt;d(C,A) = d(C,B)+d(B,A)&lt;/math&gt;

But this is exactly the definition of &lt;math&gt;[CBA]&lt;/math&gt;.

&lt;math&gt;\Box&lt;/math&gt;

(Note that the &quot;box&quot; symbol, lovingly called the &quot;Halmos&quot; in honor of [https://en.wikipedia.org/wiki/Paul_Halmos| Paul Halmos], indicates that the desired conclusion has been reached, and therefore the proof has ended.) [[File:Paul Halmos 1986.jpg|thumb|Paul Halmos, ♥]]
|}
{{robelbox/close}}
{{definition|name=line|value=
For any two distinct points, &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, define the '''line through ''A'' and ''B''''' as the set of all points satisfying any &quot;betweenness&quot; relationship with ''A'' and ''B''.  We write &lt;math&gt;\overleftrightarrow{AB}&lt;/math&gt; to denote the line, and formally define it as the set 
: &lt;math&gt;\overleftrightarrow{AB}=\{A,B\}\cup \{C\in \mathfrak P: [CAB] \text{ or } [ACB]\text{ or } [ABC]\}&lt;/math&gt;

If any two points are on the same line, we say the two points are '''collinear'''.  
}}

Read this as saying &quot;The line through ''A'' and ''B'' is defined to be the set of all points, ''C'', such that either ''A'' is between ''C'' and ''B'', or ''C'' is between ''A'' and ''B'', or ''B'' is between ''A'' and ''C''.&quot;

At this point we have finally defined all of the objects referred to in the ''Two Point Axiom''!  With all of this ground cleared, we are actually in a position to state several more axioms, and the prove another lemma.
{{robelbox|title=Three Point Axiom|theme=2}}
There exist some three distinct points which are not all collinear.
{{robelbox/close}}
{{robelbox|title=Unique Between Axiom|theme=2}}
For any three distinct collinear points, &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt;, precisely one is between the other two.
{{robelbox/close}}
{{robelbox|title= Exercise . Unique up to reordering}}
Prove that for any three distinct colinear points &lt;math&gt;A,B,C\in\mathfrak P&lt;/math&gt;, precisely one of the following holds,

* &lt;math&gt;[ABC]&lt;/math&gt;
* &lt;math&gt;[ACB]&lt;/math&gt;
* &lt;math&gt;[BAC]&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| From the ''Unique Between Axiom'' we know that precisely one of the three points is between the other two.  Let's assume ''B'' is between ''A'' and ''C'', which is to say, &lt;math&gt;[ABC]&lt;/math&gt;.

The the first bullet point holds, which already shows that at least one of the three possibilities holds.  

If &lt;math&gt;[ACB]&lt;/math&gt; were also true then ''C'' would be between ''A'' and ''B''.  But this would contract the uniqueness of the ''Unique Between Axiom'', so it is impossible.  

Likewise &lt;math&gt;[BAC]&lt;/math&gt; is impossible in the same way that we showed &lt;math&gt;[ACB]&lt;/math&gt; is impossible.  

This has now shown that if &lt;math&gt;[ABC]&lt;/math&gt; holds then the other two cannot.  ''Mutatis mutandis'', the same proof shows that if any one of the other two were true then it would prohibit the possibility of its others.  

□
|}
{{robelbox/close}}
{{robelbox|title=Infinity Axiom|theme=2}}
If &lt;math&gt;B,D\in\mathfrak P&lt;/math&gt; then there are points &lt;math&gt;A,C,E\in\mathfrak P&lt;/math&gt; such that &lt;math&gt;[ABCDE]&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise 5. Infinitely many points}}
Prove, using any of the axioms and definitions above, that there are infinitely many points.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| By the ''Three Points Axiom'' we know that there are some three points.  For now we will only need two of them, so call two of them &lt;math&gt;A_1&lt;/math&gt; and &lt;math&gt;A_2&lt;/math&gt;.

By the ''Two Points Axiom'' there is a line through these two points, &lt;math&gt;\overleftrightarrow{A_1A_2}&lt;/math&gt;.  

By the ''Infinity Axiom'' there must exist a point &lt;math&gt;A_3\in\mathfrak P&lt;/math&gt;, such that &lt;math&gt;[A_1A_2A_3]&lt;/math&gt;.  By definition of betweenness, &lt;math&gt;A_3&lt;/math&gt; must be distinct from each of &lt;math&gt;A_1,A_2&lt;/math&gt;.  So now there are three points.

Continuing likewise there must be a fourth point satisfying &lt;math&gt;[A_2A_3A_4]&lt;/math&gt;.  Continuing in the same manner, for any ''n'', there must be points &lt;math&gt;[A_1A_2\dots A_n]&lt;/math&gt;.  

Therefore, for any number ''M'', the number of points is not bounded by ''M''.  This is because there is a sequence of points &lt;math&gt;[A_1A_2\dots A_{M+1}]&lt;/math&gt;, all distinct from each other.  Then there are at least &lt;math&gt;M+1&lt;/math&gt; points and, as stated, therefore ''M'' does not bound the number of points.

Since the number of points is not bounded by any finite number, therefore it is infinite.

□
|}
{{robelbox/close}}
{{robelbox|title=Exercise 5. Distance definiteness}}
Use the axioms and definitions above to prove that, for any &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, we have 
: &lt;math&gt;d(A,B)=0&lt;/math&gt; if and only if &lt;math&gt;A=B&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| Suppose &lt;math&gt;A=B&lt;/math&gt;.  Then it follows immediately that &lt;math&gt;d(A,B) = d(A,A)&lt;/math&gt; and then by the ''Distance Axiom'' it follows that this is zero.

We now need to prove that, if &lt;math&gt;d(A,B)=0&lt;/math&gt; then therefore &lt;math&gt;A=B&lt;/math&gt;, and this will be the harder direction.  

If &lt;math&gt;d(A,B)=0&lt;/math&gt; then suppose that ''A'' is ''not B''.  Well in that case, these points are distinct, and the ''Two Point Axiom'' entails that there is a unique line through them, &lt;math&gt;\overleftrightarrow{AB}&lt;/math&gt;.  

Then by the ''Infinity Axiom'' there is a point between them, say &lt;math&gt;[ACB]&lt;/math&gt;.  Therefore these are three distinct collinear points.

As such, the ''Distance Axiom'' now tells us that 
: &lt;math&gt;d(A,B)=d(A,C)+d(C,B)&lt;/math&gt;

But since these are each nonnegative quantities, and &lt;math&gt;d(A,B)=0&lt;/math&gt; by assumption, then therefore &lt;math&gt;d(A,C)=0&lt;/math&gt; and &lt;math&gt;d(C,B)=0&lt;/math&gt;.

But now it follows that &lt;math&gt;d(A,C)=d(A,B)+d(B,C)&lt;/math&gt; using the ''Distance Axiom'' to commute some of the points in these expressions.  

But this now entails, by definition of betweenness, that &lt;math&gt;[ABC]&lt;/math&gt;.

But this shows that, of the three distinct points ''A'', ''B'', and ''C'', both ''B'' is between its other points, and also ''C'' is between its other points.

But this contradicts the ''Unique Between Axiom''.

What this shows is that, when &lt;math&gt;d(A,B)=0&lt;/math&gt;, if we further assume that &lt;math&gt;A\ne B&lt;/math&gt; then this leads to a contradiction.  Therefore we must have that &lt;math&gt;A=B&lt;/math&gt;, just as we needed to prove.  

□
|}
{{robelbox/close}}
{{robelbox|title=Exercise 6. Rays}}
Now you do it!

With inspiration taken from how we defined a line, write a reasonable definition of a ''ray''.  

Recall that, for any two distinct points &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, the ray from ''A'' through ''B'' is supposed to be effectively half of the line through ''A'' and ''B''.  In particular, it is suppose to be the half of the line which contains both of these points, and is &quot;cut&quot; at ''A''.  

This is a half-formal definition of a ray. Your challenge is to re-state this, fully formally.

Likewise, define the ''segment'' from ''A'' to ''B''.

We will write the ray from ''A'' through ''B'' by &lt;math&gt;\overrightarrow{AB}&lt;/math&gt;.  We will write the segment from ''A'' to ''B'' with &lt;math&gt;\overline{AB}&lt;/math&gt;.

Once you have defined these formally, then prove the following.  For any two distinct points &lt;math&gt;A,B\in\mathfrak P&lt;/math&gt;, 

: &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA} = \overline{AB}&lt;/math&gt;
and
: &lt;math&gt;\overrightarrow{AB}\cup\overrightarrow{BA} = \overleftrightarrow{AB}&lt;/math&gt;

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot;
| Solution
|-
| The definition of &lt;math&gt;\overrightarrow{AB}&lt;/math&gt; is the set 
: &lt;math&gt;\overrightarrow{AB}=\{C\in\mathfrak P: [ACB] \text{ or } [ABC]\}&lt;/math&gt; 
and the definition of &lt;math&gt;\overline{AB}&lt;/math&gt; is 
: &lt;math&gt;\overline{AB}=\{C\in\mathfrak P: [ACB]\}&lt;/math&gt;

Now we prove that &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA} = \overline{AB}&lt;/math&gt;.

Notice that the thing we are to prove here, is an equation ''of sets''.  Therefore, as is standard when proving the equality of two sets, we start by proofing one subset direction.

Let us start with proving &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA}\subseteq\overline{AB}&lt;/math&gt;.  In order to do so, we assume &lt;math&gt;p\in \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt; and then attempt to show &lt;math&gt;p\in\overline{AB}&lt;/math&gt;.

From &lt;math&gt; p\in\overrightarrow{AB}&lt;/math&gt; we have that either &lt;math&gt;[ApB]&lt;/math&gt; or &lt;math&gt;[ABp]&lt;/math&gt;.  From &lt;math&gt;p\in\overrightarrow{BA}&lt;/math&gt; we have that either &lt;math&gt;[BpA]&lt;/math&gt; or &lt;math&gt;[BAp]&lt;/math&gt;.

If &lt;math&gt;[ABp]&lt;/math&gt; then it would follow that we cannot have &lt;math&gt;[BpA]&lt;/math&gt; because this is equivalent (as we proved in ''The commutativity of betweenness'') to &lt;math&gt;[ApB]&lt;/math&gt;.  This, in turn, is impossible because it would contradict the ''Uniqueness Axiom''.  In turn, that is because we would have both ''B'' and ''p'' as the points which are between the others.  

By a very similar argument, ''mutatis mutandis'', if &lt;math&gt;[ABp]&lt;/math&gt; then it would follow that we cannot have &lt;math&gt;[BAp]&lt;/math&gt;.

Therefore if &lt;math&gt;[ABp]&lt;/math&gt; holds then neither &lt;math&gt;[BpA]&lt;/math&gt; nor &lt;math&gt;[BAp]&lt;/math&gt;.  But this contradicts the earlier result that we must have one or the other.  Therefore, we cannot have &lt;math&gt;[ABp]&lt;/math&gt;.

Since we established earlier that either &lt;math&gt;[ApB]&lt;/math&gt; or &lt;math&gt;[ABp]&lt;/math&gt;, but we have just seen that we cannot have &lt;math&gt;[ABp]&lt;/math&gt;, then therefore we must have &lt;math&gt;[ApB]&lt;/math&gt;.  

But then immediate from this and the definition of the segment, we must have &lt;math&gt;p\in\overline{AB}&lt;/math&gt;.

We have now shown that &lt;math&gt;\overrightarrow{AB}\cap\overrightarrow{BA}\subseteq\overline{AB}&lt;/math&gt;.  We move on to show that &lt;math&gt;\overline{AB}\subseteq\overrightarrow{AB}\cap \overrightarrow{BA}&lt;/math&gt;.

Let &lt;math&gt;p\in\overline{AB}&lt;/math&gt;, so by definition, &lt;math&gt;[ApB]&lt;/math&gt;.

Then by definition &lt;math&gt;p\in\overrightarrow{AB}&lt;/math&gt;.  By the same definition, and from the result ''The commutativity of betweenness'' exercise above, then also, &lt;math&gt;p\in\overrightarrow{BA}&lt;/math&gt;.

Therefore &lt;math&gt;p\in\overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt; as desired, and so &lt;math&gt;\overline{AB}\subseteq \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt;

This concludes the proof that &lt;math&gt;\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA}&lt;/math&gt;.  

The proof that &lt;math&gt;\overrightarrow{AB}\cup\overrightarrow{BA}=\overleftrightarrow{AB}&lt;/math&gt; is very similar.

□
|}
{{robelbox/close}}</text>
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  <page>
    <title>Studies of Boolean functions</title>
    <ns>0</ns>
    <id>304964</id>
    <revision>
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      <text bytes="1218" xml:space="preserve">These articles about Boolean functions use a similar style and terminology. &amp;nbsp;
&lt;small style=&quot;opacity: .7;&quot;&gt;They are created by [[User:Watchduck|Watchduck]] a.k.a. Tilman Piesk.&lt;/small&gt;

* '''[[Studies of Boolean functions/terminology]]'''
* [[Equivalence classes of Boolean functions]]
* [[Integer sequences related to Boolean functions]]
* [[Linear Boolean functions]]
* [[Tesseract and 16-cell faces]]

==[[Studies of Euler diagrams]]==
&lt;gallery&gt;
File:EuDi; potula (shapes).svg
File:EuDi; gilera.svg
File:EuDi; nisuke.svg
File:EuDi; kimuri.svg
File:EuDi; levana flat.svg
File:EuDi; medusa.png
File:EuDi; batch 5; 5.svg
&lt;/gallery&gt;

==Zhegalkin indices==
[[File:ANF to 1000 1001.svg|thumb|150px|left|ANF to truth table]]
[[File:Zhegalkin 256.svg|thumb|717px|right|Zhegalkin matrix]]
* [[Algebraic normal form]]
* [[Zhegalkin matrix]], [[Zhegalkin twins]]
* [[Noble Boolean functions]]
* [[Gentle sets of Boolean functions]]
* [[Linear and noble Boolean functions]]
* [[Smallest Zhegalkin index]]
{{clear}}

==old==

These old pages require some updates.

* [[3-ary Boolean functions]]
* [[4-ary Boolean functions]]
* [[Seal (discrete mathematics)]] (subgroups of nimber addition))


[[Category:Boolean functions]]</text>
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== Licensing ==
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== Licensing ==
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{{Information
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== Licensing ==
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|Date=2024-05-02
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== Licensing ==
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{{Information
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|Source={{own|Young1lim}}
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|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
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  <page>
    <title>User talk:150.176.145.117</title>
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      <comment>New resource with &quot;== May 2024 == {{subst:uw-selfrevert|Animal Farm}} ~~~~&quot;</comment>
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      <text bytes="643" xml:space="preserve">== May 2024 ==
[[File:Information.svg|25px|alt=Information icon]] Welcome to Wikiversity. Thank you for reverting your recent experiment&amp;#32;with the page [[:Animal Farm]]. Please take a look at the [[Wikiversity:Welcome|welcome page]] to learn more about contributing to our project. If you would like to experiment further, please use [[Wikiversity:Sandbox]] instead, as someone could see your edit before you revert it. Thank you. &lt;!-- Template:uw-selfrevert --&gt; &lt;!-- Template:uw-cluebotwarning1 --&gt;  [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 16:38, 2 May 2024 (UTC)</text>
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  <page>
    <title>User:Addemf/sandbox/Technical Reasoning/Logical Abstraction</title>
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      <comment>New resource with &quot;[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]  We have now seen more than a bit of axiomatic geometry.  For a lot of new students, this may seem like a challenging step up in abstraction from their previous math courses.  In this lesson,...&quot;</comment>
      <model>wikitext</model>
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      <text bytes="15670" xml:space="preserve">[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]

We have now seen more than a bit of axiomatic geometry.  For a lot of new students, this may seem like a challenging step up in abstraction from their previous math courses.

In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play.  This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

== A Case Study ==

Consider the axiom from the previous lesson, 
{{robelbox|title= Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Later on we will learn how to logically analyze an &quot;exactly one&quot; statement.  

For now let's focus instead on the simpler sentence, 
{{robelbox|title=Case study|theme=11}}
For any two points, there is a line through them.
{{robelbox/close}}

=== Abstraction ===

The goal of logic is to abstract away the specific content of any sentence, and to study the principles of reasoning which should apply to ''every'' subject.  

To abstract the sentence is to remove all of the content which is specific to geometry, while leaving the structure intact.
{{robelbox|title= Abstraction of the case study|theme=10}}
For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt;, and &lt;math&gt;T(z,x,y)&lt;/math&gt;.  
{{robelbox/close}}

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, ''x'', ''y'', and ''z''.

2. Replace the word &quot;point&quot; with the variable ''P''; and replace &quot;line&quot; with ''L''; and replace &quot;through&quot; with ''T''.  
: By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic.  The symbol ''P'' could now mean &quot;terminal&quot;, and ''L'' could mean &quot;wire&quot;, and ''T'' could mean &quot;connects&quot;.  With this specification, the abstraction would then say &quot;For every two terminals, there is a wire which connects them.&quot;

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.

: For instance, by writing &lt;math&gt;P(x)&lt;/math&gt; we indicate that object ''x'' has property ''P''.  By writing &lt;math&gt;T(z,x,y)&lt;/math&gt; we capture the idea that, whatever relationship ''T'' is, it is somehow a relationship which holds between the objects ''z'', and ''x'', and ''y''. 

Let's see one more example, this time with a specifically named object.  Take for example the sentence 

: &quot;Zero is the least number.&quot;

To abstract this sentence it will help if we first unpack some of what is meant by &quot;least&quot;.  We must have in mind the &quot;less-than-or-equal-to&quot; relationship.  Then to say that zero is the least, is to say, 

: &quot;Zero is a number and for every number zero is less-than-or-equal-to it.

With the meaning thus exposed, the sentence becomes easier to abstract.  The abstraction is,

: &lt;math&gt;N(a)&lt;/math&gt; and for every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

Notice that we use ''a'' as the abstract name of ''some'' object.  We are trying to remove the specific reference to the number zero.  

We also use the variable ''N'' for &quot;number&quot; and ''L'' for &quot;less-than-or-equal-to&quot;.
{{robelbox|title=Exercise . Abstraction exercise}}
Give the abstraction of each of the following sentences.  If any of them have the same abstraction as &quot;For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt; then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;,&quot; then indicate this.  

'' Part 1.''

Every cat is a mammal.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
|

For every ''x'', if &lt;math&gt;C(x)&lt;/math&gt; then &lt;math&gt;M(x)&lt;/math&gt;.

(''C'' abstracts &quot;cat&quot;, ''M'' abstracts &quot;mammal&quot;.)

Note: You don't have to use the same letters that I do!  The letters don't intrinsically matter, they are in a sense supposed to be &quot;meaningless&quot; because they could mean anything.
|}

''Part 2. ''

For every two numbers there is a number between them.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' and ''y'', if &lt;math&gt;N(x)&lt;/math&gt; and &lt;math&gt;N(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;N(z)&lt;/math&gt; and &lt;math&gt;B(x,z,y)&lt;/math&gt;.  

(''N'' abstracts &quot;number&quot;, ''B'' abstracts &quot;between&quot;.)

Note: You don't have to put the object names in the same order that I do!  If you wrote &lt;math&gt;B(x,y,z)&lt;/math&gt; instead, that would still be ok, as long as your abstraction shows that the three objects are in some relationship with each other.  I chose to place the ''z'' in the middle only because it is evocative of &quot;betweenness&quot; but it is not actually important to do so.
|}

''Part 3. ''

The empty set is a subset of every set.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' if &lt;math&gt;S(x)&lt;/math&gt; then &lt;math&gt;B(a,x)&lt;/math&gt;.

(''S'' abstracts &quot;set&quot;, ''B'' abstracts &quot;subset&quot;.)
|}

''Part 4.''

Every number is less than infinity.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

(''N'' abstracts &quot;number&quot;, ''L'' abstracts &quot;less-than&quot;, ''a'' abstracts &quot;infinity&quot;.)
|}

''Part 5.''

For any two natural numbers, there is a rational number between them.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot; style=&quot;width: 100%&quot;
| Solution
|- 
| The abstraction of this sentence is exactly like the case study.  Therefore, up to a different choice of letters, its abstraction should look the same.
|}

{{robelbox/close}}
{{robelbox|title= Exercise . Specification}}
To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be ''anything''.  

The reverse of abstraction is specification.  That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act &quot;specification&quot;.  

Consider the sentence, which we will call ''S'', below.

: ''S'' is the sentence &quot;2 is even and positive.&quot;

Below is the abstraction of ''S''.

: ''A'' is the abstraction &quot;''t'' is &lt;math&gt;E(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

The specification which takes us from ''A'' back to ''S'' can be expressed by a function &lt;math&gt;\sigma&lt;/math&gt; which maps each variable to a word.  In particular &lt;math&gt;\sigma(t) = 2&lt;/math&gt; and &lt;math&gt; \sigma(E)&lt;/math&gt; = &quot;even&quot; and &lt;math&gt;\sigma(P) &lt;/math&gt; = &quot;positive&quot;.  

Therefore applying this specification &lt;math&gt;\sigma&lt;/math&gt; to abstraction ''A'' results in sentence ''S''.  

Let us now define a different specification, &lt;math&gt;\tau&lt;/math&gt;.  Define this to be the function &lt;math&gt;\tau(t)&lt;/math&gt; = Biden and &lt;math&gt;\tau(E)&lt;/math&gt; = &quot;president&quot;, and &lt;math&gt;\tau(P)&lt;/math&gt; = &quot;Democrat&quot;.  

What is the sentence which results from applying specification &lt;math&gt;\tau&lt;/math&gt; to abstraction ''A''?  
{{robelbox/close}}
{{robelbox|title=Exercise . One to one or many}}
True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.
{{robelbox/close}}

== Symbolization ==

=== Conjunction ===

It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators.  The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples.  To begin with, consider the (abstract) sentence &quot;&lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;&quot;.  

To symbolize this we would merely replace the conjunction &quot;and&quot; with the symbol &lt;math&gt;\land&lt;/math&gt;.  Therefore its symbolization is 

: &lt;math&gt;P(x)\land P(y)&lt;/math&gt;

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction &quot;conjuncts&quot;.  Therefore &lt;math&gt;P(x)&lt;/math&gt; is the left conjunct and &lt;math&gt;P(y)&lt;/math&gt; is the right conjunct.  

=== Disjunction ===

Consider the example sentence &quot;6 is divisible by 3 or 4.&quot;  Its abstraction is

: &quot;&lt;math&gt;D(s,t)&lt;/math&gt; or &lt;math&gt;D(s,f)&lt;/math&gt;.&quot;

where ''s'' abstracts 6, ''t'' abstracts 3, ''f'' abstracts 4, and ''D'' abstracts &quot;divisible by&quot;.

We use the symbol &lt;math&gt;\lor&lt;/math&gt; in place of the word &quot;or&quot; so that the symbolization of this is 

: &lt;math&gt;D(s,t)\lor D(s,f)&lt;/math&gt;

Because of the use of &quot;or&quot; this sentence is a disjunction, and we call each clause a &quot;disjunct&quot;.  

=== Negation ===

Consider next the sentence &quot;The cat is not on the mat,&quot; with abstraction 

: &quot;Not &lt;math&gt;O(c,m)&lt;/math&gt;&quot;

where ''c'' abstracts the cat, ''m'' abstracts the mat, and ''O'' abstracts the &quot;is on&quot; relation.  

We symbolize this by 

: &lt;math&gt;\neg O(c,m)&lt;/math&gt;

Note that negation is a unary operator, unlike &lt;math&gt;\land&lt;/math&gt; and &lt;math&gt;\lor&lt;/math&gt; which are binary operators.  Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for &quot;the clause under the negation&quot; the way that there are words for conjuncts and disjuncts.  However, if we would like a word, we might choose the Latin conjugation ''negationem''.  (Literally: the thing negated.)

=== Conditional ===

Consider the next sentence &quot;If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed.&quot;   This is abstracted as

: &quot;&lt;math&gt;P(y, n, f)&lt;/math&gt; then &lt;math&gt;T(c)&lt;/math&gt;&quot;

where ''y'' abstracts you, ''n'' abstracts 9 a.m., ''f'' abstracts 5 p.m., ''c'' abstracts your car, ''P'' abstracts the &quot;parks&quot; relation, and ''T'' abstracts the &quot;is towed&quot; relation.  

This is symbolized as 

: &lt;math&gt;P(y,n,f)\to T(c)&lt;/math&gt;

It is worth appreciating how sometimes the alignment of English and symbolization is awkward.  In English we often indicate the condition with the word &quot;if&quot; and then signal the consequence with &quot;then&quot;.  

However, in symbols we only have a single &quot;infix&quot; symbol.  We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.  

Up to now, I've been describing the clauses of a conditional as &quot;condition&quot; and &quot;consequence&quot;.  However, the more technical terms which is used by logicians are &quot;antecedent&quot; and &quot;consequent&quot;.  

Therefore, from now on, we will use the more correct vocabulary.  In an expression of the form &quot;''P'' \to ''Q''&quot;, we will say that ''P'' is the antecedent and ''Q'' the consequent.

=== Universal Quantification ===

The last two symbols that we will study are the &quot;quantifiers&quot;.  

Consider the (false) sentence &quot;Every number divisible by 2 is divisible by 4.&quot;  Its abstraction is 

: &quot;For every ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;.&quot;

where ''t'' abstracts 2, ''f'' abstracts 4, and ''D'' abstracts the &quot;divides&quot; relation.  

Naturally &quot;if (condition) then (consequence)&quot; portion of this can be symbolized with &lt;math&gt;\to&lt;/math&gt;.  But what about the &quot;for every ''x''&quot;?

Notice that the propositional operators above, &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt; all took some number of sentences, and used them to form a new sentence.  But that is not what the &quot;for all ''x''&quot; part of this sentence does.  

Rather, &quot;for all ''x''&quot; takes a so-called &quot;open formula&quot; and turns it into a sentence.  

An open formula is something like ''x = x'', which is technically not a sentence because it gives you no indicate of ''what x is''.  

But if we attach this to the universal quantifier, and say &quot;For all ''x'', we have ''x = x'',&quot; this now becomes a sentence because we are told what ''x'' means.  In particular, when we write this, ''x'' represents any arbitrary object.  And this makes the sentence a claim about ''all'' objects in the universe.  

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then &quot;For all ''x'', we have ''x = x''&quot; would entail that 1 = 1.  If 1/2 is in the universe (and for most mathematical conversations, it is) then it would ''also'' entail 1/2 = 1/2.  And so on.  

Because quantification plays a significantly different role than propositional connectives, we give it different notation.  The sentence &quot;For all ''x'', we have ''x = x'',&quot; has symbolization 

: &lt;math&gt;\forall x(x=x)&lt;/math&gt;

The symbol &lt;math&gt;\forall &lt;/math&gt; is the universal quantifier, read as &quot;for all&quot;, or &quot;for any&quot;, whichever the speaker prefers.  It is always written with a variable, called the variable of its quantification.  

After that we write an open formula, which is some sentence which uses ''x''.  In this example the open formula is &quot;''x = x''&quot;.  

To return to the earlier example, &quot;For all ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;&quot;, this has symbolization 

: &lt;math&gt;\forall x(D(x,t)\to D(x,f))&lt;/math&gt;

Notice the importance of the parentheses wrapping around the open formula &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  If we only wrote 

: &lt;math&gt;\forall x D(x,t)\to D(x,f)&lt;/math&gt;

then it would be ambiguous whether the &lt;math&gt;\forall x&lt;/math&gt; is meant to apply only to &lt;math&gt;D(x,t)&lt;/math&gt; or the entire &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  

(The reader might be able to reasonably guess the intended reading.  But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

=== Existential Quantification ===

Consider the (true) sentence &quot;There is an even prime number.&quot;  This has abstraction 

: &quot;There is an ''x'' such that &lt;math&gt;E(x)&lt;/math&gt; and &lt;math&gt;P(x)&lt;/math&gt;.&quot;

We use the symbol &lt;math&gt;\exists&lt;/math&gt; for the phrase &quot;There is&quot; or &quot;There exists&quot;.  Therefore this sentence has symbolization  

: &lt;math&gt;\exists x (E(x)\land P(x))&lt;/math&gt;

Existential quantification, like universal quantification, works at the level of objects.  But universal quantification requires that ''whichever'' object you assign to the variable ''x'', you always get a true sentence.  The existential quantifier only requires that there is at least one such object.  

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.  

== Conclusion of the Case Study ==

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Sentence Symbolization.gif|400px|The reading of an abstracted sentence into a symbolic sentence.]]
|}

Finally we may symbolize the sentence with which we started this lesson.  Recall that the sentence was 

: &quot;For every two points there is a line through them.&quot;

It had abstraction 

: For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;.

Notice that, this time, there are two &quot;for all&quot; variables. Therefore we need two quantifiers for the initial choice of ''x'' and ''y''.  

Moreover, there is an existential sentence &quot;in the middle&quot; of the conditional statement.  

The rest of the symbolization should be readable at this point, and so we present it below.

: &lt;math&gt;\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)&lt;/math&gt;

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.</text>
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      <parentid>2624773</parentid>
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      <text bytes="16372" xml:space="preserve">[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]

In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play.  This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

== A Case Study ==

Consider the axiom from the previous lesson, 
{{robelbox|title= Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Later on we will learn how to logically analyze an &quot;exactly one&quot; statement.  

For now let's focus instead on the simpler sentence, 
{{robelbox|title=Case study|theme=11}}
For any two points, there is a line through them.
{{robelbox/close}}

=== Abstraction ===

The goal of logic is to abstract away the specific content of any sentence.  

Thereby, in logic, we study the principles of reasoning which should apply to ''every'' subject.  

To abstract the &quot;case study&quot; sentence above, is to remove all of the content which is specific to geometry, while leaving the structure intact.
{{robelbox|title= Abstraction of the case study|theme=10}}
For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt;, and &lt;math&gt;T(z,x,y)&lt;/math&gt;.  
{{robelbox/close}}

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, ''x'', ''y'', and ''z''.

2. Replace the word &quot;point&quot; with the variable ''P''; and replace &quot;line&quot; with ''L''; and replace &quot;through&quot; with ''T''.  

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.

: For instance, by writing &lt;math&gt;P(x)&lt;/math&gt; we indicate that object ''x'' has property ''P''.  By writing &lt;math&gt;T(z,x,y)&lt;/math&gt; we capture the idea that, whatever relationship ''T'' is, it is somehow a relationship which holds between the objects ''z'', and ''x'', and ''y''. 

=== Specification ===

By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic.  The symbol ''P'' could now mean anything.  

For example, ''P'' could mean &quot;terminal&quot;, and ''L'' could mean &quot;wire&quot;, and ''T'' could mean &quot;connects&quot;.  Whereas &quot;abstraction&quot; is the removal of details, what we are discussing here is taking an abstraction and ''supplying'' details.  We will call this &quot;specification&quot;.  

With the specification &lt;math&gt;[P\to \text{terminal}, L\to \text{wire}, T\to \text{connects}]&lt;/math&gt;, applied to the abstraction, results in the sentence 
: &quot;For every two terminals, there is a wire which connects them.&quot;

=== A Second Case Study ===

Let's see one more example, this time with a specifically named object.  Take for example the sentence 

: &quot;Zero is the least number.&quot;

In this example, the sentence specifically refers to one &quot;named&quot; object, which is the number zero.

Before we try to abstract this sentence, it will help if we first unpack some of what is meant by &quot;least&quot;.  We must have in mind the &quot;less-than-or-equal-to&quot; relationship.  

Then to say that zero is the least, is to say, 

: &quot;Zero is a number. And zero is less-than-or-equal-to every number.&quot;

With the meaning thus exposed, the sentence becomes easier to abstract.  The abstraction is,

: &lt;math&gt;N(a)&lt;/math&gt; and for every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

''N'' is the abstraction of &quot;number&quot;.  ''L'' is the abstraction of &quot;less-than-or-equal-to&quot;.  ''a'' is the abstraction of &quot;zero&quot;.  

Note that we tend to use letters ''a'' through ''t'' as abstract symbols for specifically named objects.  You can think of these as &quot;proper names&quot;.  

By contrast we use letters like ''x'', ''y'', and ''z'' for variables.  In the abstraction above, ''x'' is a variable because it could refer to anything.  We will also use ''t'' through ''w'' for variables if needed, although we have a slight (and meaningless) preference for ''x'', ''y'', and ''z''.  
{{robelbox|title=Exercise . Abstraction exercise}}
Give the abstraction of each of the following sentences.  If any of them have the same abstraction as &quot;For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt; then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;,&quot; then indicate this.  

'' Part 1.''

Every cat is a mammal.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
|

For every ''x'', if &lt;math&gt;C(x)&lt;/math&gt; then &lt;math&gt;M(x)&lt;/math&gt;.

(''C'' abstracts &quot;cat&quot;, ''M'' abstracts &quot;mammal&quot;.)

Note: You don't have to use the same letters that I do!  The letters don't intrinsically matter, they are in a sense supposed to be &quot;meaningless&quot; because they could mean anything.
|}

''Part 2. ''

For every two numbers there is a number between them.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' and ''y'', if &lt;math&gt;N(x)&lt;/math&gt; and &lt;math&gt;N(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;N(z)&lt;/math&gt; and &lt;math&gt;B(x,z,y)&lt;/math&gt;.  

(''N'' abstracts &quot;number&quot;, ''B'' abstracts &quot;between&quot;.)

Note: You don't have to put the object names in the same order that I do!  If you wrote &lt;math&gt;B(x,y,z)&lt;/math&gt; instead, that would still be ok, as long as your abstraction shows that the three objects are in some relationship with each other.  I chose to place the ''z'' in the middle only because it is evocative of &quot;betweenness&quot; but it is not actually important to do so.
|}

''Part 3. ''

The empty set is a subset of every set.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' if &lt;math&gt;S(x)&lt;/math&gt; then &lt;math&gt;B(a,x)&lt;/math&gt;.

(''S'' abstracts &quot;set&quot;, ''B'' abstracts &quot;subset&quot;.)
|}

''Part 4.''

Every number is less than infinity.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

(''N'' abstracts &quot;number&quot;, ''L'' abstracts &quot;less-than&quot;, ''a'' abstracts &quot;infinity&quot;.)
|}

''Part 5.''

For any two natural numbers, there is a rational number between them.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot; style=&quot;width: 100%&quot;
| Solution
|- 
| The abstraction of this sentence is exactly like the case study.  Therefore, up to a different choice of letters, its abstraction should look the same.
|}

{{robelbox/close}}
{{robelbox|title= Exercise . Specification}}
To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be ''anything''.  

The reverse of abstraction is specification.  That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act &quot;specification&quot;.  

Consider the sentence, which we will call ''S'', below.

: ''S'' is the sentence &quot;2 is even and positive.&quot;

Below is the abstraction of ''S''.

: ''A'' is the abstraction &quot;''t'' is &lt;math&gt;E(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

The specification which takes us from ''A'' back to ''S'' can be expressed by a function &lt;math&gt;\sigma&lt;/math&gt; which maps each variable to a word.  In particular &lt;math&gt;\sigma(t) = 2&lt;/math&gt; and &lt;math&gt; \sigma(E)&lt;/math&gt; = &quot;even&quot; and &lt;math&gt;\sigma(P) &lt;/math&gt; = &quot;positive&quot;.  

Therefore applying this specification &lt;math&gt;\sigma&lt;/math&gt; to abstraction ''A'' results in sentence ''S''.  

Let us now define a different specification, &lt;math&gt;\tau&lt;/math&gt;.  Define this to be the function &lt;math&gt;\tau(t)&lt;/math&gt; = Biden and &lt;math&gt;\tau(E)&lt;/math&gt; = &quot;president&quot;, and &lt;math&gt;\tau(P)&lt;/math&gt; = &quot;Democrat&quot;.  

What is the sentence which results from applying specification &lt;math&gt;\tau&lt;/math&gt; to abstraction ''A''?  
{{robelbox/close}}
{{robelbox|title=Exercise . One to one or many}}
True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.
{{robelbox/close}}

== Symbolization ==

=== Conjunction ===

It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators.  The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples.  To begin with, consider the (abstract) sentence &quot;&lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;&quot;.  

To symbolize this we would merely replace the conjunction &quot;and&quot; with the symbol &lt;math&gt;\land&lt;/math&gt;.  Therefore its symbolization is 

: &lt;math&gt;P(x)\land P(y)&lt;/math&gt;

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction &quot;conjuncts&quot;.  Therefore &lt;math&gt;P(x)&lt;/math&gt; is the left conjunct and &lt;math&gt;P(y)&lt;/math&gt; is the right conjunct.  

=== Disjunction ===

Consider the example sentence &quot;6 is divisible by 3 or 4.&quot;  Its abstraction is

: &quot;&lt;math&gt;D(s,t)&lt;/math&gt; or &lt;math&gt;D(s,f)&lt;/math&gt;.&quot;

where ''s'' abstracts 6, ''t'' abstracts 3, ''f'' abstracts 4, and ''D'' abstracts &quot;divisible by&quot;.

We use the symbol &lt;math&gt;\lor&lt;/math&gt; in place of the word &quot;or&quot; so that the symbolization of this is 

: &lt;math&gt;D(s,t)\lor D(s,f)&lt;/math&gt;

Because of the use of &quot;or&quot; this sentence is a disjunction, and we call each clause a &quot;disjunct&quot;.  

=== Negation ===

Consider next the sentence &quot;The cat is not on the mat,&quot; with abstraction 

: &quot;Not &lt;math&gt;O(c,m)&lt;/math&gt;&quot;

where ''c'' abstracts the cat, ''m'' abstracts the mat, and ''O'' abstracts the &quot;is on&quot; relation.  

We symbolize this by 

: &lt;math&gt;\neg O(c,m)&lt;/math&gt;

Note that negation is a unary operator, unlike &lt;math&gt;\land&lt;/math&gt; and &lt;math&gt;\lor&lt;/math&gt; which are binary operators.  Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for &quot;the clause under the negation&quot; the way that there are words for conjuncts and disjuncts.  However, if we would like a word, we might choose the Latin conjugation ''negationem''.  (Literally: the thing negated.)

=== Conditional ===

Consider the next sentence &quot;If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed.&quot;   This is abstracted as

: &quot;&lt;math&gt;P(y, n, f)&lt;/math&gt; then &lt;math&gt;T(c)&lt;/math&gt;&quot;

where ''y'' abstracts you, ''n'' abstracts 9 a.m., ''f'' abstracts 5 p.m., ''c'' abstracts your car, ''P'' abstracts the &quot;parks&quot; relation, and ''T'' abstracts the &quot;is towed&quot; relation.  

This is symbolized as 

: &lt;math&gt;P(y,n,f)\to T(c)&lt;/math&gt;

It is worth appreciating how sometimes the alignment of English and symbolization is awkward.  In English we often indicate the condition with the word &quot;if&quot; and then signal the consequence with &quot;then&quot;.  

However, in symbols we only have a single &quot;infix&quot; symbol.  We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.  

Up to now, I've been describing the clauses of a conditional as &quot;condition&quot; and &quot;consequence&quot;.  However, the more technical terms which is used by logicians are &quot;antecedent&quot; and &quot;consequent&quot;.  

Therefore, from now on, we will use the more correct vocabulary.  In an expression of the form &quot;''P'' \to ''Q''&quot;, we will say that ''P'' is the antecedent and ''Q'' the consequent.

=== Universal Quantification ===

The last two symbols that we will study are the &quot;quantifiers&quot;.  

Consider the (false) sentence &quot;Every number divisible by 2 is divisible by 4.&quot;  Its abstraction is 

: &quot;For every ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;.&quot;

where ''t'' abstracts 2, ''f'' abstracts 4, and ''D'' abstracts the &quot;divides&quot; relation.  

Naturally &quot;if (condition) then (consequence)&quot; portion of this can be symbolized with &lt;math&gt;\to&lt;/math&gt;.  But what about the &quot;for every ''x''&quot;?

Notice that the propositional operators above, &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt; all took some number of sentences, and used them to form a new sentence.  But that is not what the &quot;for all ''x''&quot; part of this sentence does.  

Rather, &quot;for all ''x''&quot; takes a so-called &quot;open formula&quot; and turns it into a sentence.  

An open formula is something like ''x = x'', which is technically not a sentence because it gives you no indicate of ''what x is''.  

But if we attach this to the universal quantifier, and say &quot;For all ''x'', we have ''x = x'',&quot; this now becomes a sentence because we are told what ''x'' means.  In particular, when we write this, ''x'' represents any arbitrary object.  And this makes the sentence a claim about ''all'' objects in the universe.  

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then &quot;For all ''x'', we have ''x = x''&quot; would entail that 1 = 1.  If 1/2 is in the universe (and for most mathematical conversations, it is) then it would ''also'' entail 1/2 = 1/2.  And so on.  

Because quantification plays a significantly different role than propositional connectives, we give it different notation.  The sentence &quot;For all ''x'', we have ''x = x'',&quot; has symbolization 

: &lt;math&gt;\forall x(x=x)&lt;/math&gt;

The symbol &lt;math&gt;\forall &lt;/math&gt; is the universal quantifier, read as &quot;for all&quot;, or &quot;for any&quot;, whichever the speaker prefers.  It is always written with a variable, called the variable of its quantification.  

After that we write an open formula, which is some sentence which uses ''x''.  In this example the open formula is &quot;''x = x''&quot;.  

To return to the earlier example, &quot;For all ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;&quot;, this has symbolization 

: &lt;math&gt;\forall x(D(x,t)\to D(x,f))&lt;/math&gt;

Notice the importance of the parentheses wrapping around the open formula &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  If we only wrote 

: &lt;math&gt;\forall x D(x,t)\to D(x,f)&lt;/math&gt;

then it would be ambiguous whether the &lt;math&gt;\forall x&lt;/math&gt; is meant to apply only to &lt;math&gt;D(x,t)&lt;/math&gt; or the entire &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  

(The reader might be able to reasonably guess the intended reading.  But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

=== Existential Quantification ===

Consider the (true) sentence &quot;There is an even prime number.&quot;  This has abstraction 

: &quot;There is an ''x'' such that &lt;math&gt;E(x)&lt;/math&gt; and &lt;math&gt;P(x)&lt;/math&gt;.&quot;

We use the symbol &lt;math&gt;\exists&lt;/math&gt; for the phrase &quot;There is&quot; or &quot;There exists&quot;.  Therefore this sentence has symbolization  

: &lt;math&gt;\exists x (E(x)\land P(x))&lt;/math&gt;

Existential quantification, like universal quantification, works at the level of objects.  But universal quantification requires that ''whichever'' object you assign to the variable ''x'', you always get a true sentence.  The existential quantifier only requires that there is at least one such object.  

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.  

== Conclusion of the Case Study ==

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Sentence Symbolization.gif|400px|The reading of an abstracted sentence into a symbolic sentence.]]
|}

Finally we may symbolize the sentence with which we started this lesson.  Recall that the sentence was 

: &quot;For every two points there is a line through them.&quot;

It had abstraction 

: For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;.

Notice that, this time, there are two &quot;for all&quot; variables. Therefore we need two quantifiers for the initial choice of ''x'' and ''y''.  

Moreover, there is an existential sentence &quot;in the middle&quot; of the conditional statement.  

The rest of the symbolization should be readable at this point, and so we present it below.

: &lt;math&gt;\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)&lt;/math&gt;

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.</text>
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    </revision>
    <revision>
      <id>2624833</id>
      <parentid>2624774</parentid>
      <timestamp>2024-05-02T19:29:38Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Specification */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16384" xml:space="preserve">[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]

In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play.  This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

== A Case Study ==

Consider the axiom from the previous lesson, 
{{robelbox|title= Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Later on we will learn how to logically analyze an &quot;exactly one&quot; statement.  

For now let's focus instead on the simpler sentence, 
{{robelbox|title=Case study|theme=11}}
For any two points, there is a line through them.
{{robelbox/close}}

=== Abstraction ===

The goal of logic is to abstract away the specific content of any sentence.  

Thereby, in logic, we study the principles of reasoning which should apply to ''every'' subject.  

To abstract the &quot;case study&quot; sentence above, is to remove all of the content which is specific to geometry, while leaving the structure intact.
{{robelbox|title= Abstraction of the case study|theme=10}}
For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt;, and &lt;math&gt;T(z,x,y)&lt;/math&gt;.  
{{robelbox/close}}

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, ''x'', ''y'', and ''z''.

2. Replace the word &quot;point&quot; with the variable ''P''; and replace &quot;line&quot; with ''L''; and replace &quot;through&quot; with ''T''.  

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.

: For instance, by writing &lt;math&gt;P(x)&lt;/math&gt; we indicate that object ''x'' has property ''P''.  By writing &lt;math&gt;T(z,x,y)&lt;/math&gt; we capture the idea that, whatever relationship ''T'' is, it is somehow a relationship which holds between the objects ''z'', and ''x'', and ''y''. 

=== Specification ===

By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic.  The symbol ''P'' could now mean anything.  

For example, ''P'' could mean &quot;terminal&quot;, and ''L'' could mean &quot;wire&quot;, and ''T'' could mean &quot;connects&quot;.  

Whereas &quot;abstraction&quot; is the removal of details, what we are discussing here is taking an abstraction and ''supplying'' details.  We will call this &quot;specification&quot;.  

The specification 
: &lt;math&gt;[\![ P\to \text{terminal}, L\to \text{wire}, T\to \text{connects}]\!]&lt;/math&gt;
applied to the abstraction above, results in the sentence 
: &quot;For every two terminals, there is a wire which connects them.&quot;

=== A Second Case Study ===

Let's see one more example, this time with a specifically named object.  Take for example the sentence 

: &quot;Zero is the least number.&quot;

In this example, the sentence specifically refers to one &quot;named&quot; object, which is the number zero.

Before we try to abstract this sentence, it will help if we first unpack some of what is meant by &quot;least&quot;.  We must have in mind the &quot;less-than-or-equal-to&quot; relationship.  

Then to say that zero is the least, is to say, 

: &quot;Zero is a number. And zero is less-than-or-equal-to every number.&quot;

With the meaning thus exposed, the sentence becomes easier to abstract.  The abstraction is,

: &lt;math&gt;N(a)&lt;/math&gt; and for every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

''N'' is the abstraction of &quot;number&quot;.  ''L'' is the abstraction of &quot;less-than-or-equal-to&quot;.  ''a'' is the abstraction of &quot;zero&quot;.  

Note that we tend to use letters ''a'' through ''t'' as abstract symbols for specifically named objects.  You can think of these as &quot;proper names&quot;.  

By contrast we use letters like ''x'', ''y'', and ''z'' for variables.  In the abstraction above, ''x'' is a variable because it could refer to anything.  We will also use ''t'' through ''w'' for variables if needed, although we have a slight (and meaningless) preference for ''x'', ''y'', and ''z''.  
{{robelbox|title=Exercise . Abstraction exercise}}
Give the abstraction of each of the following sentences.  If any of them have the same abstraction as &quot;For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt; then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;,&quot; then indicate this.  

'' Part 1.''

Every cat is a mammal.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
|

For every ''x'', if &lt;math&gt;C(x)&lt;/math&gt; then &lt;math&gt;M(x)&lt;/math&gt;.

(''C'' abstracts &quot;cat&quot;, ''M'' abstracts &quot;mammal&quot;.)

Note: You don't have to use the same letters that I do!  The letters don't intrinsically matter, they are in a sense supposed to be &quot;meaningless&quot; because they could mean anything.
|}

''Part 2. ''

For every two numbers there is a number between them.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' and ''y'', if &lt;math&gt;N(x)&lt;/math&gt; and &lt;math&gt;N(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;N(z)&lt;/math&gt; and &lt;math&gt;B(x,z,y)&lt;/math&gt;.  

(''N'' abstracts &quot;number&quot;, ''B'' abstracts &quot;between&quot;.)

Note: You don't have to put the object names in the same order that I do!  If you wrote &lt;math&gt;B(x,y,z)&lt;/math&gt; instead, that would still be ok, as long as your abstraction shows that the three objects are in some relationship with each other.  I chose to place the ''z'' in the middle only because it is evocative of &quot;betweenness&quot; but it is not actually important to do so.
|}

''Part 3. ''

The empty set is a subset of every set.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' if &lt;math&gt;S(x)&lt;/math&gt; then &lt;math&gt;B(a,x)&lt;/math&gt;.

(''S'' abstracts &quot;set&quot;, ''B'' abstracts &quot;subset&quot;.)
|}

''Part 4.''

Every number is less than infinity.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

(''N'' abstracts &quot;number&quot;, ''L'' abstracts &quot;less-than&quot;, ''a'' abstracts &quot;infinity&quot;.)
|}

''Part 5.''

For any two natural numbers, there is a rational number between them.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot; style=&quot;width: 100%&quot;
| Solution
|- 
| The abstraction of this sentence is exactly like the case study.  Therefore, up to a different choice of letters, its abstraction should look the same.
|}

{{robelbox/close}}
{{robelbox|title= Exercise . Specification}}
To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be ''anything''.  

The reverse of abstraction is specification.  That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act &quot;specification&quot;.  

Consider the sentence, which we will call ''S'', below.

: ''S'' is the sentence &quot;2 is even and positive.&quot;

Below is the abstraction of ''S''.

: ''A'' is the abstraction &quot;''t'' is &lt;math&gt;E(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

The specification which takes us from ''A'' back to ''S'' can be expressed by a function &lt;math&gt;\sigma&lt;/math&gt; which maps each variable to a word.  In particular &lt;math&gt;\sigma(t) = 2&lt;/math&gt; and &lt;math&gt; \sigma(E)&lt;/math&gt; = &quot;even&quot; and &lt;math&gt;\sigma(P) &lt;/math&gt; = &quot;positive&quot;.  

Therefore applying this specification &lt;math&gt;\sigma&lt;/math&gt; to abstraction ''A'' results in sentence ''S''.  

Let us now define a different specification, &lt;math&gt;\tau&lt;/math&gt;.  Define this to be the function &lt;math&gt;\tau(t)&lt;/math&gt; = Biden and &lt;math&gt;\tau(E)&lt;/math&gt; = &quot;president&quot;, and &lt;math&gt;\tau(P)&lt;/math&gt; = &quot;Democrat&quot;.  

What is the sentence which results from applying specification &lt;math&gt;\tau&lt;/math&gt; to abstraction ''A''?  
{{robelbox/close}}
{{robelbox|title=Exercise . One to one or many}}
True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.
{{robelbox/close}}

== Symbolization ==

=== Conjunction ===

It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators.  The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples.  To begin with, consider the (abstract) sentence &quot;&lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;&quot;.  

To symbolize this we would merely replace the conjunction &quot;and&quot; with the symbol &lt;math&gt;\land&lt;/math&gt;.  Therefore its symbolization is 

: &lt;math&gt;P(x)\land P(y)&lt;/math&gt;

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction &quot;conjuncts&quot;.  Therefore &lt;math&gt;P(x)&lt;/math&gt; is the left conjunct and &lt;math&gt;P(y)&lt;/math&gt; is the right conjunct.  

=== Disjunction ===

Consider the example sentence &quot;6 is divisible by 3 or 4.&quot;  Its abstraction is

: &quot;&lt;math&gt;D(s,t)&lt;/math&gt; or &lt;math&gt;D(s,f)&lt;/math&gt;.&quot;

where ''s'' abstracts 6, ''t'' abstracts 3, ''f'' abstracts 4, and ''D'' abstracts &quot;divisible by&quot;.

We use the symbol &lt;math&gt;\lor&lt;/math&gt; in place of the word &quot;or&quot; so that the symbolization of this is 

: &lt;math&gt;D(s,t)\lor D(s,f)&lt;/math&gt;

Because of the use of &quot;or&quot; this sentence is a disjunction, and we call each clause a &quot;disjunct&quot;.  

=== Negation ===

Consider next the sentence &quot;The cat is not on the mat,&quot; with abstraction 

: &quot;Not &lt;math&gt;O(c,m)&lt;/math&gt;&quot;

where ''c'' abstracts the cat, ''m'' abstracts the mat, and ''O'' abstracts the &quot;is on&quot; relation.  

We symbolize this by 

: &lt;math&gt;\neg O(c,m)&lt;/math&gt;

Note that negation is a unary operator, unlike &lt;math&gt;\land&lt;/math&gt; and &lt;math&gt;\lor&lt;/math&gt; which are binary operators.  Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for &quot;the clause under the negation&quot; the way that there are words for conjuncts and disjuncts.  However, if we would like a word, we might choose the Latin conjugation ''negationem''.  (Literally: the thing negated.)

=== Conditional ===

Consider the next sentence &quot;If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed.&quot;   This is abstracted as

: &quot;&lt;math&gt;P(y, n, f)&lt;/math&gt; then &lt;math&gt;T(c)&lt;/math&gt;&quot;

where ''y'' abstracts you, ''n'' abstracts 9 a.m., ''f'' abstracts 5 p.m., ''c'' abstracts your car, ''P'' abstracts the &quot;parks&quot; relation, and ''T'' abstracts the &quot;is towed&quot; relation.  

This is symbolized as 

: &lt;math&gt;P(y,n,f)\to T(c)&lt;/math&gt;

It is worth appreciating how sometimes the alignment of English and symbolization is awkward.  In English we often indicate the condition with the word &quot;if&quot; and then signal the consequence with &quot;then&quot;.  

However, in symbols we only have a single &quot;infix&quot; symbol.  We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.  

Up to now, I've been describing the clauses of a conditional as &quot;condition&quot; and &quot;consequence&quot;.  However, the more technical terms which is used by logicians are &quot;antecedent&quot; and &quot;consequent&quot;.  

Therefore, from now on, we will use the more correct vocabulary.  In an expression of the form &quot;''P'' \to ''Q''&quot;, we will say that ''P'' is the antecedent and ''Q'' the consequent.

=== Universal Quantification ===

The last two symbols that we will study are the &quot;quantifiers&quot;.  

Consider the (false) sentence &quot;Every number divisible by 2 is divisible by 4.&quot;  Its abstraction is 

: &quot;For every ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;.&quot;

where ''t'' abstracts 2, ''f'' abstracts 4, and ''D'' abstracts the &quot;divides&quot; relation.  

Naturally &quot;if (condition) then (consequence)&quot; portion of this can be symbolized with &lt;math&gt;\to&lt;/math&gt;.  But what about the &quot;for every ''x''&quot;?

Notice that the propositional operators above, &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt; all took some number of sentences, and used them to form a new sentence.  But that is not what the &quot;for all ''x''&quot; part of this sentence does.  

Rather, &quot;for all ''x''&quot; takes a so-called &quot;open formula&quot; and turns it into a sentence.  

An open formula is something like ''x = x'', which is technically not a sentence because it gives you no indicate of ''what x is''.  

But if we attach this to the universal quantifier, and say &quot;For all ''x'', we have ''x = x'',&quot; this now becomes a sentence because we are told what ''x'' means.  In particular, when we write this, ''x'' represents any arbitrary object.  And this makes the sentence a claim about ''all'' objects in the universe.  

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then &quot;For all ''x'', we have ''x = x''&quot; would entail that 1 = 1.  If 1/2 is in the universe (and for most mathematical conversations, it is) then it would ''also'' entail 1/2 = 1/2.  And so on.  

Because quantification plays a significantly different role than propositional connectives, we give it different notation.  The sentence &quot;For all ''x'', we have ''x = x'',&quot; has symbolization 

: &lt;math&gt;\forall x(x=x)&lt;/math&gt;

The symbol &lt;math&gt;\forall &lt;/math&gt; is the universal quantifier, read as &quot;for all&quot;, or &quot;for any&quot;, whichever the speaker prefers.  It is always written with a variable, called the variable of its quantification.  

After that we write an open formula, which is some sentence which uses ''x''.  In this example the open formula is &quot;''x = x''&quot;.  

To return to the earlier example, &quot;For all ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;&quot;, this has symbolization 

: &lt;math&gt;\forall x(D(x,t)\to D(x,f))&lt;/math&gt;

Notice the importance of the parentheses wrapping around the open formula &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  If we only wrote 

: &lt;math&gt;\forall x D(x,t)\to D(x,f)&lt;/math&gt;

then it would be ambiguous whether the &lt;math&gt;\forall x&lt;/math&gt; is meant to apply only to &lt;math&gt;D(x,t)&lt;/math&gt; or the entire &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  

(The reader might be able to reasonably guess the intended reading.  But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

=== Existential Quantification ===

Consider the (true) sentence &quot;There is an even prime number.&quot;  This has abstraction 

: &quot;There is an ''x'' such that &lt;math&gt;E(x)&lt;/math&gt; and &lt;math&gt;P(x)&lt;/math&gt;.&quot;

We use the symbol &lt;math&gt;\exists&lt;/math&gt; for the phrase &quot;There is&quot; or &quot;There exists&quot;.  Therefore this sentence has symbolization  

: &lt;math&gt;\exists x (E(x)\land P(x))&lt;/math&gt;

Existential quantification, like universal quantification, works at the level of objects.  But universal quantification requires that ''whichever'' object you assign to the variable ''x'', you always get a true sentence.  The existential quantifier only requires that there is at least one such object.  

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.  

== Conclusion of the Case Study ==

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Sentence Symbolization.gif|400px|The reading of an abstracted sentence into a symbolic sentence.]]
|}

Finally we may symbolize the sentence with which we started this lesson.  Recall that the sentence was 

: &quot;For every two points there is a line through them.&quot;

It had abstraction 

: For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;.

Notice that, this time, there are two &quot;for all&quot; variables. Therefore we need two quantifiers for the initial choice of ''x'' and ''y''.  

Moreover, there is an existential sentence &quot;in the middle&quot; of the conditional statement.  

The rest of the symbolization should be readable at this point, and so we present it below.

: &lt;math&gt;\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)&lt;/math&gt;

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.</text>
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    </revision>
    <revision>
      <id>2624834</id>
      <parentid>2624833</parentid>
      <timestamp>2024-05-02T19:31:25Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* A Second Case Study */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16379" xml:space="preserve">[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]

In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play.  This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

== A Case Study ==

Consider the axiom from the previous lesson, 
{{robelbox|title= Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Later on we will learn how to logically analyze an &quot;exactly one&quot; statement.  

For now let's focus instead on the simpler sentence, 
{{robelbox|title=Case study|theme=11}}
For any two points, there is a line through them.
{{robelbox/close}}

=== Abstraction ===

The goal of logic is to abstract away the specific content of any sentence.  

Thereby, in logic, we study the principles of reasoning which should apply to ''every'' subject.  

To abstract the &quot;case study&quot; sentence above, is to remove all of the content which is specific to geometry, while leaving the structure intact.
{{robelbox|title= Abstraction of the case study|theme=10}}
For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt;, and &lt;math&gt;T(z,x,y)&lt;/math&gt;.  
{{robelbox/close}}

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, ''x'', ''y'', and ''z''.

2. Replace the word &quot;point&quot; with the variable ''P''; and replace &quot;line&quot; with ''L''; and replace &quot;through&quot; with ''T''.  

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.

: For instance, by writing &lt;math&gt;P(x)&lt;/math&gt; we indicate that object ''x'' has property ''P''.  By writing &lt;math&gt;T(z,x,y)&lt;/math&gt; we capture the idea that, whatever relationship ''T'' is, it is somehow a relationship which holds between the objects ''z'', and ''x'', and ''y''. 

=== Specification ===

By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic.  The symbol ''P'' could now mean anything.  

For example, ''P'' could mean &quot;terminal&quot;, and ''L'' could mean &quot;wire&quot;, and ''T'' could mean &quot;connects&quot;.  

Whereas &quot;abstraction&quot; is the removal of details, what we are discussing here is taking an abstraction and ''supplying'' details.  We will call this &quot;specification&quot;.  

The specification 
: &lt;math&gt;[\![ P\to \text{terminal}, L\to \text{wire}, T\to \text{connects}]\!]&lt;/math&gt;
applied to the abstraction above, results in the sentence 
: &quot;For every two terminals, there is a wire which connects them.&quot;

=== A Second Case Study ===

Let's see one more example, this time with a specifically named object.  Take for example the sentence 

: &quot;Zero is the least number.&quot;

In this example, the sentence specifically refers to one &quot;named&quot; object, which is the number zero.

Before we try to abstract this sentence, it will help if we first unpack some of what is meant by &quot;least&quot;.  We must have in mind the &quot;less-than-or-equal-to&quot; relationship.  

To say that zero is the least, is to say, 

: &quot;Zero is a number. And zero is less-than-or-equal-to every number.&quot;

With the meaning thus exposed, the sentence becomes easier to abstract.  The abstraction is,

: &lt;math&gt;N(a)&lt;/math&gt; and for every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

''N'' is the abstraction of &quot;number&quot;.  ''L'' is the abstraction of &quot;less-than-or-equal-to&quot;.  ''a'' is the abstraction of &quot;zero&quot;.  

Note that we tend to use letters ''a'' through ''t'' as abstract symbols for specifically named objects.  You can think of these as &quot;proper names&quot;.  

By contrast we use letters like ''x'', ''y'', and ''z'' for variables.  In the abstraction above, ''x'' is a variable because it could refer to anything.  We will also use ''t'' through ''w'' for variables if needed, although we have a slight (and meaningless) preference for ''x'', ''y'', and ''z''.  
{{robelbox|title=Exercise . Abstraction exercise}}
Give the abstraction of each of the following sentences.  If any of them have the same abstraction as &quot;For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt; then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;,&quot; then indicate this.  

'' Part 1.''

Every cat is a mammal.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
|

For every ''x'', if &lt;math&gt;C(x)&lt;/math&gt; then &lt;math&gt;M(x)&lt;/math&gt;.

(''C'' abstracts &quot;cat&quot;, ''M'' abstracts &quot;mammal&quot;.)

Note: You don't have to use the same letters that I do!  The letters don't intrinsically matter, they are in a sense supposed to be &quot;meaningless&quot; because they could mean anything.
|}

''Part 2. ''

For every two numbers there is a number between them.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' and ''y'', if &lt;math&gt;N(x)&lt;/math&gt; and &lt;math&gt;N(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;N(z)&lt;/math&gt; and &lt;math&gt;B(x,z,y)&lt;/math&gt;.  

(''N'' abstracts &quot;number&quot;, ''B'' abstracts &quot;between&quot;.)

Note: You don't have to put the object names in the same order that I do!  If you wrote &lt;math&gt;B(x,y,z)&lt;/math&gt; instead, that would still be ok, as long as your abstraction shows that the three objects are in some relationship with each other.  I chose to place the ''z'' in the middle only because it is evocative of &quot;betweenness&quot; but it is not actually important to do so.
|}

''Part 3. ''

The empty set is a subset of every set.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' if &lt;math&gt;S(x)&lt;/math&gt; then &lt;math&gt;B(a,x)&lt;/math&gt;.

(''S'' abstracts &quot;set&quot;, ''B'' abstracts &quot;subset&quot;.)
|}

''Part 4.''

Every number is less than infinity.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

(''N'' abstracts &quot;number&quot;, ''L'' abstracts &quot;less-than&quot;, ''a'' abstracts &quot;infinity&quot;.)
|}

''Part 5.''

For any two natural numbers, there is a rational number between them.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot; style=&quot;width: 100%&quot;
| Solution
|- 
| The abstraction of this sentence is exactly like the case study.  Therefore, up to a different choice of letters, its abstraction should look the same.
|}

{{robelbox/close}}
{{robelbox|title= Exercise . Specification}}
To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be ''anything''.  

The reverse of abstraction is specification.  That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act &quot;specification&quot;.  

Consider the sentence, which we will call ''S'', below.

: ''S'' is the sentence &quot;2 is even and positive.&quot;

Below is the abstraction of ''S''.

: ''A'' is the abstraction &quot;''t'' is &lt;math&gt;E(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

The specification which takes us from ''A'' back to ''S'' can be expressed by a function &lt;math&gt;\sigma&lt;/math&gt; which maps each variable to a word.  In particular &lt;math&gt;\sigma(t) = 2&lt;/math&gt; and &lt;math&gt; \sigma(E)&lt;/math&gt; = &quot;even&quot; and &lt;math&gt;\sigma(P) &lt;/math&gt; = &quot;positive&quot;.  

Therefore applying this specification &lt;math&gt;\sigma&lt;/math&gt; to abstraction ''A'' results in sentence ''S''.  

Let us now define a different specification, &lt;math&gt;\tau&lt;/math&gt;.  Define this to be the function &lt;math&gt;\tau(t)&lt;/math&gt; = Biden and &lt;math&gt;\tau(E)&lt;/math&gt; = &quot;president&quot;, and &lt;math&gt;\tau(P)&lt;/math&gt; = &quot;Democrat&quot;.  

What is the sentence which results from applying specification &lt;math&gt;\tau&lt;/math&gt; to abstraction ''A''?  
{{robelbox/close}}
{{robelbox|title=Exercise . One to one or many}}
True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.
{{robelbox/close}}

== Symbolization ==

=== Conjunction ===

It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators.  The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples.  To begin with, consider the (abstract) sentence &quot;&lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;&quot;.  

To symbolize this we would merely replace the conjunction &quot;and&quot; with the symbol &lt;math&gt;\land&lt;/math&gt;.  Therefore its symbolization is 

: &lt;math&gt;P(x)\land P(y)&lt;/math&gt;

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction &quot;conjuncts&quot;.  Therefore &lt;math&gt;P(x)&lt;/math&gt; is the left conjunct and &lt;math&gt;P(y)&lt;/math&gt; is the right conjunct.  

=== Disjunction ===

Consider the example sentence &quot;6 is divisible by 3 or 4.&quot;  Its abstraction is

: &quot;&lt;math&gt;D(s,t)&lt;/math&gt; or &lt;math&gt;D(s,f)&lt;/math&gt;.&quot;

where ''s'' abstracts 6, ''t'' abstracts 3, ''f'' abstracts 4, and ''D'' abstracts &quot;divisible by&quot;.

We use the symbol &lt;math&gt;\lor&lt;/math&gt; in place of the word &quot;or&quot; so that the symbolization of this is 

: &lt;math&gt;D(s,t)\lor D(s,f)&lt;/math&gt;

Because of the use of &quot;or&quot; this sentence is a disjunction, and we call each clause a &quot;disjunct&quot;.  

=== Negation ===

Consider next the sentence &quot;The cat is not on the mat,&quot; with abstraction 

: &quot;Not &lt;math&gt;O(c,m)&lt;/math&gt;&quot;

where ''c'' abstracts the cat, ''m'' abstracts the mat, and ''O'' abstracts the &quot;is on&quot; relation.  

We symbolize this by 

: &lt;math&gt;\neg O(c,m)&lt;/math&gt;

Note that negation is a unary operator, unlike &lt;math&gt;\land&lt;/math&gt; and &lt;math&gt;\lor&lt;/math&gt; which are binary operators.  Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for &quot;the clause under the negation&quot; the way that there are words for conjuncts and disjuncts.  However, if we would like a word, we might choose the Latin conjugation ''negationem''.  (Literally: the thing negated.)

=== Conditional ===

Consider the next sentence &quot;If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed.&quot;   This is abstracted as

: &quot;&lt;math&gt;P(y, n, f)&lt;/math&gt; then &lt;math&gt;T(c)&lt;/math&gt;&quot;

where ''y'' abstracts you, ''n'' abstracts 9 a.m., ''f'' abstracts 5 p.m., ''c'' abstracts your car, ''P'' abstracts the &quot;parks&quot; relation, and ''T'' abstracts the &quot;is towed&quot; relation.  

This is symbolized as 

: &lt;math&gt;P(y,n,f)\to T(c)&lt;/math&gt;

It is worth appreciating how sometimes the alignment of English and symbolization is awkward.  In English we often indicate the condition with the word &quot;if&quot; and then signal the consequence with &quot;then&quot;.  

However, in symbols we only have a single &quot;infix&quot; symbol.  We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.  

Up to now, I've been describing the clauses of a conditional as &quot;condition&quot; and &quot;consequence&quot;.  However, the more technical terms which is used by logicians are &quot;antecedent&quot; and &quot;consequent&quot;.  

Therefore, from now on, we will use the more correct vocabulary.  In an expression of the form &quot;''P'' \to ''Q''&quot;, we will say that ''P'' is the antecedent and ''Q'' the consequent.

=== Universal Quantification ===

The last two symbols that we will study are the &quot;quantifiers&quot;.  

Consider the (false) sentence &quot;Every number divisible by 2 is divisible by 4.&quot;  Its abstraction is 

: &quot;For every ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;.&quot;

where ''t'' abstracts 2, ''f'' abstracts 4, and ''D'' abstracts the &quot;divides&quot; relation.  

Naturally &quot;if (condition) then (consequence)&quot; portion of this can be symbolized with &lt;math&gt;\to&lt;/math&gt;.  But what about the &quot;for every ''x''&quot;?

Notice that the propositional operators above, &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt; all took some number of sentences, and used them to form a new sentence.  But that is not what the &quot;for all ''x''&quot; part of this sentence does.  

Rather, &quot;for all ''x''&quot; takes a so-called &quot;open formula&quot; and turns it into a sentence.  

An open formula is something like ''x = x'', which is technically not a sentence because it gives you no indicate of ''what x is''.  

But if we attach this to the universal quantifier, and say &quot;For all ''x'', we have ''x = x'',&quot; this now becomes a sentence because we are told what ''x'' means.  In particular, when we write this, ''x'' represents any arbitrary object.  And this makes the sentence a claim about ''all'' objects in the universe.  

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then &quot;For all ''x'', we have ''x = x''&quot; would entail that 1 = 1.  If 1/2 is in the universe (and for most mathematical conversations, it is) then it would ''also'' entail 1/2 = 1/2.  And so on.  

Because quantification plays a significantly different role than propositional connectives, we give it different notation.  The sentence &quot;For all ''x'', we have ''x = x'',&quot; has symbolization 

: &lt;math&gt;\forall x(x=x)&lt;/math&gt;

The symbol &lt;math&gt;\forall &lt;/math&gt; is the universal quantifier, read as &quot;for all&quot;, or &quot;for any&quot;, whichever the speaker prefers.  It is always written with a variable, called the variable of its quantification.  

After that we write an open formula, which is some sentence which uses ''x''.  In this example the open formula is &quot;''x = x''&quot;.  

To return to the earlier example, &quot;For all ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;&quot;, this has symbolization 

: &lt;math&gt;\forall x(D(x,t)\to D(x,f))&lt;/math&gt;

Notice the importance of the parentheses wrapping around the open formula &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  If we only wrote 

: &lt;math&gt;\forall x D(x,t)\to D(x,f)&lt;/math&gt;

then it would be ambiguous whether the &lt;math&gt;\forall x&lt;/math&gt; is meant to apply only to &lt;math&gt;D(x,t)&lt;/math&gt; or the entire &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  

(The reader might be able to reasonably guess the intended reading.  But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

=== Existential Quantification ===

Consider the (true) sentence &quot;There is an even prime number.&quot;  This has abstraction 

: &quot;There is an ''x'' such that &lt;math&gt;E(x)&lt;/math&gt; and &lt;math&gt;P(x)&lt;/math&gt;.&quot;

We use the symbol &lt;math&gt;\exists&lt;/math&gt; for the phrase &quot;There is&quot; or &quot;There exists&quot;.  Therefore this sentence has symbolization  

: &lt;math&gt;\exists x (E(x)\land P(x))&lt;/math&gt;

Existential quantification, like universal quantification, works at the level of objects.  But universal quantification requires that ''whichever'' object you assign to the variable ''x'', you always get a true sentence.  The existential quantifier only requires that there is at least one such object.  

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.  

== Conclusion of the Case Study ==

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Sentence Symbolization.gif|400px|The reading of an abstracted sentence into a symbolic sentence.]]
|}

Finally we may symbolize the sentence with which we started this lesson.  Recall that the sentence was 

: &quot;For every two points there is a line through them.&quot;

It had abstraction 

: For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;.

Notice that, this time, there are two &quot;for all&quot; variables. Therefore we need two quantifiers for the initial choice of ''x'' and ''y''.  

Moreover, there is an existential sentence &quot;in the middle&quot; of the conditional statement.  

The rest of the symbolization should be readable at this point, and so we present it below.

: &lt;math&gt;\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)&lt;/math&gt;

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.</text>
      <sha1>8b1tv501ka53jzx94yh8gk661acwo0q</sha1>
    </revision>
    <revision>
      <id>2624835</id>
      <parentid>2624834</parentid>
      <timestamp>2024-05-02T19:41:10Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* A Second Case Study */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16524" xml:space="preserve">[[File:Abstraction_-_Davis.jpg|thumb|Abstract art.  [https://en.wikipedia.org/wiki/Abstraction| Abstraction] is generally the elimination of details, in order to reason and talk about things at a &quot;[https://en.wikipedia.org/wiki/High-_and_low-level| higher level] of description&quot;.  ]]

In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play.  This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

== A Case Study ==

Consider the axiom from the previous lesson, 
{{robelbox|title= Two Point Axiom|theme=2}}
For any two points there is exactly one line through them.  
{{robelbox/close}}

Later on we will learn how to logically analyze an &quot;exactly one&quot; statement.  

For now let's focus instead on the simpler sentence, 
{{robelbox|title=Case study|theme=11}}
For any two points, there is a line through them.
{{robelbox/close}}

=== Abstraction ===

The goal of logic is to abstract away the specific content of any sentence.  

Thereby, in logic, we study the principles of reasoning which should apply to ''every'' subject.  

To abstract the &quot;case study&quot; sentence above, is to remove all of the content which is specific to geometry, while leaving the structure intact.
{{robelbox|title= Abstraction of the case study|theme=10}}
For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt;, and &lt;math&gt;T(z,x,y)&lt;/math&gt;.  
{{robelbox/close}}

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, ''x'', ''y'', and ''z''.

2. Replace the word &quot;point&quot; with the variable ''P''; and replace &quot;line&quot; with ''L''; and replace &quot;through&quot; with ''T''.  

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.

: For instance, by writing &lt;math&gt;P(x)&lt;/math&gt; we indicate that object ''x'' has property ''P''.  By writing &lt;math&gt;T(z,x,y)&lt;/math&gt; we capture the idea that, whatever relationship ''T'' is, it is somehow a relationship which holds between the objects ''z'', and ''x'', and ''y''. 

=== Specification ===

By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic.  The symbol ''P'' could now mean anything.  

For example, ''P'' could mean &quot;terminal&quot;, and ''L'' could mean &quot;wire&quot;, and ''T'' could mean &quot;connects&quot;.  

Whereas &quot;abstraction&quot; is the removal of details, what we are discussing here is taking an abstraction and ''supplying'' details.  We will call this &quot;specification&quot;.  

The specification 
: &lt;math&gt;[\![ P\to \text{terminal}, L\to \text{wire}, T\to \text{connects}]\!]&lt;/math&gt;
applied to the abstraction above, results in the sentence 
: &quot;For every two terminals, there is a wire which connects them.&quot;

=== A Second Case Study ===

Let's see one more example, this time with a specifically named object.  Take for example the sentence 

: &quot;Zero is the least number.&quot;

In this example, the sentence specifically refers to one &quot;named&quot; object, which is the number zero.

Before we try to abstract this sentence, it will help if we first unpack some of what is meant by &quot;least&quot;.  We must have in mind the &quot;less-than-or-equal-to&quot; relationship.  

To say that zero is the least number, is to say, 

: &quot;Zero is a number. And zero is less-than-or-equal-to every number.&quot;

But it is even better to write the sentence as 

: &quot;Zero is a number.  And for every ''x'', if ''x'' is a number then zero is less-than-or-equal-to ''x''.&quot;

This second version is much easier to abstract, and the abstraction is,

: &lt;math&gt;N(a)&lt;/math&gt;.  And for every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

''N'' is the abstraction of &quot;number&quot;.  ''L'' is the abstraction of &quot;less-than-or-equal-to&quot;.  ''a'' is the abstraction of &quot;zero&quot;.  

Note that we tend to use letters ''a'' through ''t'' as abstract symbols for specifically named objects.  You can think of these as &quot;proper names&quot;.  

By contrast we use letters like ''x'', ''y'', and ''z'' for variables.  In the abstraction above, ''x'' is a variable because it could refer to anything.  We will also use ''t'' through ''w'' for variables if needed, although we have a slight (and meaningless) preference for ''x'', ''y'', and ''z''.  
{{robelbox|title=Exercise . Abstraction exercise}}
Give the abstraction of each of the following sentences.  If any of them have the same abstraction as &quot;For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt; then there is a ''z'' which is &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;,&quot; then indicate this.  

'' Part 1.''

Every cat is a mammal.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
|

For every ''x'', if &lt;math&gt;C(x)&lt;/math&gt; then &lt;math&gt;M(x)&lt;/math&gt;.

(''C'' abstracts &quot;cat&quot;, ''M'' abstracts &quot;mammal&quot;.)

Note: You don't have to use the same letters that I do!  The letters don't intrinsically matter, they are in a sense supposed to be &quot;meaningless&quot; because they could mean anything.
|}

''Part 2. ''

For every two numbers there is a number between them.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' and ''y'', if &lt;math&gt;N(x)&lt;/math&gt; and &lt;math&gt;N(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;N(z)&lt;/math&gt; and &lt;math&gt;B(x,z,y)&lt;/math&gt;.  

(''N'' abstracts &quot;number&quot;, ''B'' abstracts &quot;between&quot;.)

Note: You don't have to put the object names in the same order that I do!  If you wrote &lt;math&gt;B(x,y,z)&lt;/math&gt; instead, that would still be ok, as long as your abstraction shows that the three objects are in some relationship with each other.  I chose to place the ''z'' in the middle only because it is evocative of &quot;betweenness&quot; but it is not actually important to do so.
|}

''Part 3. ''

The empty set is a subset of every set.  

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'' if &lt;math&gt;S(x)&lt;/math&gt; then &lt;math&gt;B(a,x)&lt;/math&gt;.

(''S'' abstracts &quot;set&quot;, ''B'' abstracts &quot;subset&quot;.)
|}

''Part 4.''

Every number is less than infinity.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible mw-collapsed&quot; style=&quot;width: 100%&quot;
| Solution
|-
| For every ''x'', if &lt;math&gt;N(x)&lt;/math&gt; then &lt;math&gt;L(a,x)&lt;/math&gt;.

(''N'' abstracts &quot;number&quot;, ''L'' abstracts &quot;less-than&quot;, ''a'' abstracts &quot;infinity&quot;.)
|}

''Part 5.''

For any two natural numbers, there is a rational number between them.

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible&quot; style=&quot;width: 100%&quot;
| Solution
|- 
| The abstraction of this sentence is exactly like the case study.  Therefore, up to a different choice of letters, its abstraction should look the same.
|}

{{robelbox/close}}
{{robelbox|title= Exercise . Specification}}
To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be ''anything''.  

The reverse of abstraction is specification.  That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act &quot;specification&quot;.  

Consider the sentence, which we will call ''S'', below.

: ''S'' is the sentence &quot;2 is even and positive.&quot;

Below is the abstraction of ''S''.

: ''A'' is the abstraction &quot;''t'' is &lt;math&gt;E(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

The specification which takes us from ''A'' back to ''S'' can be expressed by a function &lt;math&gt;\sigma&lt;/math&gt; which maps each variable to a word.  In particular &lt;math&gt;\sigma(t) = 2&lt;/math&gt; and &lt;math&gt; \sigma(E)&lt;/math&gt; = &quot;even&quot; and &lt;math&gt;\sigma(P) &lt;/math&gt; = &quot;positive&quot;.  

Therefore applying this specification &lt;math&gt;\sigma&lt;/math&gt; to abstraction ''A'' results in sentence ''S''.  

Let us now define a different specification, &lt;math&gt;\tau&lt;/math&gt;.  Define this to be the function &lt;math&gt;\tau(t)&lt;/math&gt; = Biden and &lt;math&gt;\tau(E)&lt;/math&gt; = &quot;president&quot;, and &lt;math&gt;\tau(P)&lt;/math&gt; = &quot;Democrat&quot;.  

What is the sentence which results from applying specification &lt;math&gt;\tau&lt;/math&gt; to abstraction ''A''?  
{{robelbox/close}}
{{robelbox|title=Exercise . One to one or many}}
True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.
{{robelbox/close}}

== Symbolization ==

=== Conjunction ===

It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators.  The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples.  To begin with, consider the (abstract) sentence &quot;&lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;&quot;.  

To symbolize this we would merely replace the conjunction &quot;and&quot; with the symbol &lt;math&gt;\land&lt;/math&gt;.  Therefore its symbolization is 

: &lt;math&gt;P(x)\land P(y)&lt;/math&gt;

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction &quot;conjuncts&quot;.  Therefore &lt;math&gt;P(x)&lt;/math&gt; is the left conjunct and &lt;math&gt;P(y)&lt;/math&gt; is the right conjunct.  

=== Disjunction ===

Consider the example sentence &quot;6 is divisible by 3 or 4.&quot;  Its abstraction is

: &quot;&lt;math&gt;D(s,t)&lt;/math&gt; or &lt;math&gt;D(s,f)&lt;/math&gt;.&quot;

where ''s'' abstracts 6, ''t'' abstracts 3, ''f'' abstracts 4, and ''D'' abstracts &quot;divisible by&quot;.

We use the symbol &lt;math&gt;\lor&lt;/math&gt; in place of the word &quot;or&quot; so that the symbolization of this is 

: &lt;math&gt;D(s,t)\lor D(s,f)&lt;/math&gt;

Because of the use of &quot;or&quot; this sentence is a disjunction, and we call each clause a &quot;disjunct&quot;.  

=== Negation ===

Consider next the sentence &quot;The cat is not on the mat,&quot; with abstraction 

: &quot;Not &lt;math&gt;O(c,m)&lt;/math&gt;&quot;

where ''c'' abstracts the cat, ''m'' abstracts the mat, and ''O'' abstracts the &quot;is on&quot; relation.  

We symbolize this by 

: &lt;math&gt;\neg O(c,m)&lt;/math&gt;

Note that negation is a unary operator, unlike &lt;math&gt;\land&lt;/math&gt; and &lt;math&gt;\lor&lt;/math&gt; which are binary operators.  Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for &quot;the clause under the negation&quot; the way that there are words for conjuncts and disjuncts.  However, if we would like a word, we might choose the Latin conjugation ''negationem''.  (Literally: the thing negated.)

=== Conditional ===

Consider the next sentence &quot;If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed.&quot;   This is abstracted as

: &quot;&lt;math&gt;P(y, n, f)&lt;/math&gt; then &lt;math&gt;T(c)&lt;/math&gt;&quot;

where ''y'' abstracts you, ''n'' abstracts 9 a.m., ''f'' abstracts 5 p.m., ''c'' abstracts your car, ''P'' abstracts the &quot;parks&quot; relation, and ''T'' abstracts the &quot;is towed&quot; relation.  

This is symbolized as 

: &lt;math&gt;P(y,n,f)\to T(c)&lt;/math&gt;

It is worth appreciating how sometimes the alignment of English and symbolization is awkward.  In English we often indicate the condition with the word &quot;if&quot; and then signal the consequence with &quot;then&quot;.  

However, in symbols we only have a single &quot;infix&quot; symbol.  We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.  

Up to now, I've been describing the clauses of a conditional as &quot;condition&quot; and &quot;consequence&quot;.  However, the more technical terms which is used by logicians are &quot;antecedent&quot; and &quot;consequent&quot;.  

Therefore, from now on, we will use the more correct vocabulary.  In an expression of the form &quot;''P'' \to ''Q''&quot;, we will say that ''P'' is the antecedent and ''Q'' the consequent.

=== Universal Quantification ===

The last two symbols that we will study are the &quot;quantifiers&quot;.  

Consider the (false) sentence &quot;Every number divisible by 2 is divisible by 4.&quot;  Its abstraction is 

: &quot;For every ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;.&quot;

where ''t'' abstracts 2, ''f'' abstracts 4, and ''D'' abstracts the &quot;divides&quot; relation.  

Naturally &quot;if (condition) then (consequence)&quot; portion of this can be symbolized with &lt;math&gt;\to&lt;/math&gt;.  But what about the &quot;for every ''x''&quot;?

Notice that the propositional operators above, &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt; all took some number of sentences, and used them to form a new sentence.  But that is not what the &quot;for all ''x''&quot; part of this sentence does.  

Rather, &quot;for all ''x''&quot; takes a so-called &quot;open formula&quot; and turns it into a sentence.  

An open formula is something like ''x = x'', which is technically not a sentence because it gives you no indicate of ''what x is''.  

But if we attach this to the universal quantifier, and say &quot;For all ''x'', we have ''x = x'',&quot; this now becomes a sentence because we are told what ''x'' means.  In particular, when we write this, ''x'' represents any arbitrary object.  And this makes the sentence a claim about ''all'' objects in the universe.  

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then &quot;For all ''x'', we have ''x = x''&quot; would entail that 1 = 1.  If 1/2 is in the universe (and for most mathematical conversations, it is) then it would ''also'' entail 1/2 = 1/2.  And so on.  

Because quantification plays a significantly different role than propositional connectives, we give it different notation.  The sentence &quot;For all ''x'', we have ''x = x'',&quot; has symbolization 

: &lt;math&gt;\forall x(x=x)&lt;/math&gt;

The symbol &lt;math&gt;\forall &lt;/math&gt; is the universal quantifier, read as &quot;for all&quot;, or &quot;for any&quot;, whichever the speaker prefers.  It is always written with a variable, called the variable of its quantification.  

After that we write an open formula, which is some sentence which uses ''x''.  In this example the open formula is &quot;''x = x''&quot;.  

To return to the earlier example, &quot;For all ''x'', if &lt;math&gt;D(x,t)&lt;/math&gt; then &lt;math&gt;D(x,f)&lt;/math&gt;&quot;, this has symbolization 

: &lt;math&gt;\forall x(D(x,t)\to D(x,f))&lt;/math&gt;

Notice the importance of the parentheses wrapping around the open formula &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  If we only wrote 

: &lt;math&gt;\forall x D(x,t)\to D(x,f)&lt;/math&gt;

then it would be ambiguous whether the &lt;math&gt;\forall x&lt;/math&gt; is meant to apply only to &lt;math&gt;D(x,t)&lt;/math&gt; or the entire &lt;math&gt;D(x,t)\to D(x,f)&lt;/math&gt;.  

(The reader might be able to reasonably guess the intended reading.  But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

=== Existential Quantification ===

Consider the (true) sentence &quot;There is an even prime number.&quot;  This has abstraction 

: &quot;There is an ''x'' such that &lt;math&gt;E(x)&lt;/math&gt; and &lt;math&gt;P(x)&lt;/math&gt;.&quot;

We use the symbol &lt;math&gt;\exists&lt;/math&gt; for the phrase &quot;There is&quot; or &quot;There exists&quot;.  Therefore this sentence has symbolization  

: &lt;math&gt;\exists x (E(x)\land P(x))&lt;/math&gt;

Existential quantification, like universal quantification, works at the level of objects.  But universal quantification requires that ''whichever'' object you assign to the variable ''x'', you always get a true sentence.  The existential quantifier only requires that there is at least one such object.  

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.  

== Conclusion of the Case Study ==

{| role=&quot;presentation&quot; class=&quot;wikitable mw-collapsible floatright&quot;
|
|-
| [[File:Sentence Symbolization.gif|400px|The reading of an abstracted sentence into a symbolic sentence.]]
|}

Finally we may symbolize the sentence with which we started this lesson.  Recall that the sentence was 

: &quot;For every two points there is a line through them.&quot;

It had abstraction 

: For every ''x'' and ''y'', if &lt;math&gt;P(x)&lt;/math&gt; and &lt;math&gt;P(y)&lt;/math&gt;, then there is a ''z'' such that &lt;math&gt;L(z)&lt;/math&gt; and &lt;math&gt;T(z,x,y)&lt;/math&gt;.

Notice that, this time, there are two &quot;for all&quot; variables. Therefore we need two quantifiers for the initial choice of ''x'' and ''y''.  

Moreover, there is an existential sentence &quot;in the middle&quot; of the conditional statement.  

The rest of the symbolization should be readable at this point, and so we present it below.

: &lt;math&gt;\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)&lt;/math&gt;

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.</text>
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  <page>
    <title>User:Addemf/sandbox/Technical Reasoning/Propositional Expressions</title>
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      <id>2624839</id>
      <timestamp>2024-05-02T20:30:08Z</timestamp>
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        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>New resource with &quot;Now that we've seen the basic idea of abstraction and symbolization, we build a formal system around the idea.    To begin with, because it is simpler, we focus for now on logic at the level of propositions — so-called &quot;propositional logic&quot;.  This means that we do not focus on the quantifiers from the previous lesson, but we do focus on the operators like &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt;.  File:Semantic Modeling.png|thumb|A cartoon depiction of a database (right) whi...&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27864" xml:space="preserve">Now that we've seen the basic idea of abstraction and symbolization, we build a formal system around the idea.  

To begin with, because it is simpler, we focus for now on logic at the level of propositions — so-called &quot;propositional logic&quot;.  This means that we do not focus on the quantifiers from the previous lesson, but we do focus on the operators like &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt;.

[[File:Semantic Modeling.png|thumb|A cartoon depiction of a database (right) which holds some data or software objects.  These data are supposed to be in correspondence with things in the real world (left), like buildings or trees or mountains.  This serves as a metaphor for the relationship between syntax and semantics in logic.]]

The most important theme throughout this lesson, is the relationship between ''syntax'' and ''semantics''.  

Syntax is the &quot;dead&quot;, mechanical, &quot;calculational&quot; system of writing some symbols followed (according to a formal rule) by certain other symbols.  To take one example, everything that a computer ''does'' is syntactic.

Semantics is the study of the meaning, which gives those dead symbols life, so to speak.  A computer might display symbols, simply because it was programmed to do so.  But it is the human mind, reading those symbols, which ''gives'' the symbols their meaning.  

== Propositional Variables ==

=== Syntax ===

Because our current interest is in propositions, we will no longer write expressions like ''P(x, y, z)'' as we did in the previous lesson, since we will have nothing to say about the objects ''x, y,'' and ''z''.  We will return to object variables when we get to the lessons on &quot;first-order logic&quot;.   

We will instead use propositional variables, like 

: ''P, Q, R, ...'', 

usually denoted by capital italics Roman letters.  

From a syntactic point of view, these will be the basic &quot;building blocks&quot; or &quot;atoms&quot; which build up to more complex expressions.  

=== Semantics ===

The formal definition of exactly what a propositional variable is, would probably be too technical and cause more confusion than insight at this point.  Therefore let us just rely on the intuition that a propositional variable is an abstraction of any proposition.  

Propositions may be true or false.  

Moreover, they must be precisely one of these two options.  

Therefore the abstraction of a proposition, is essentially the fact that it may be true or false.  

We represent this fact in a so-called &quot;truth table&quot;, like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
| ''P''
|-
| T
|-
| F
|}

The top row shows a propositional variable, and below it, the possible &quot;truth-values&quot; that it could take.  These are true and false, annotated as &quot;T&quot; and &quot;F&quot; respectively.  

At this moment, that might seem like a needlessly elaborate display of a simple idea.  However, things are about to get complicated soon, and it will be valuable to have tables to organize our information.

=== Caveats ===

Note that the semantics we are giving here are the semantics of mathematics, but they are not exactly the same as the semantics of everyday speech.  

For one thing, the semantics of everyday speech includes much more than propositions.  

But also, there are some philosophical disputes about exactly what the semantics of statements are.  One could make nonsensical statements, like &quot;green is round&quot;, which seems not just false but meaningless.  In general, [https://en.wikipedia.org/wiki/Category_mistake| category mistakes] of all kinds often seem neither true nor false but meaningless.

Another example is &quot;the present king of France is bald.&quot;  There is (as of writing this) no present kind of France.  Some philosophers argue that this sentence is technically false (see [https://en.wikipedia.org/wiki/Theory_of_descriptions| the theory of descriptions]), but many philosophers and linguists reject this analysis.  However, there is no consensus on how to regard such a sentence.  

One could go on, but let's not get distracted.  Just be aware that the semantics presented here are those used in mathematical settings, and that is a very useful setting for analyzing a wide range of real-world phenomena.  But these semantics are narrowly limited.

== Negation ==

=== Syntax ===

The negation of a propositional variable, like ''P'', is written as &lt;math&gt;\neg P&lt;/math&gt;.  

Notice that although ''P'' is a propositional variable, &lt;math&gt;\neg P&lt;/math&gt; is not.  Rather, it is an example of what we will eventually call a ''propositional formula''.  We are not yet in a position to define this phrase exactly; but toward the end of this lesson, we will.

A propositional formula is essentially just anything that is built from propositional variables using propositional operators (negation, conjunction, disjunction, and conditional).  

Moreover, notice that we may negate ''any'' propositional formula, not just variables.   Therefore 

: &lt;math&gt;\neg (\neg P)&lt;/math&gt;
and 
: &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;

are propositional formulas too.  

Note the use of parentheses, which will become increasingly important as the complexity of propositional formulas grows.  We may write &lt;math&gt;(\neg P)&lt;/math&gt; if we wanted to, or &lt;math&gt;(\neg(\neg P))&lt;/math&gt;.  But we also allow ourselves to omit the outer-most parentheses.  

=== Semantics ===

The semantics of negation is to switch true to false, and false to true.  

Therefore we write the truth table of negation like so 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! &lt;math&gt;\neg P&lt;/math&gt;
|-
| F
|-
| T
|}

One often wants to juxtapose this, with the truth-table from which the negated values were derived, to help see what happened here.  

Therefore we prefer to write the truth-table instead like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot; 
|-
! |''P'' 
!style=&quot;background-color: lightblue;&quot; |
! |&lt;math&gt;\neg &lt;/math&gt; 
! |&lt;math&gt;P&lt;/math&gt;
|-
| T|| style=&quot;background-color: lightblue;&quot; |
| F|| T
|-
| F|| style=&quot;background-color: lightblue;&quot;|
|T||F
|}

The left-most column simply records the original arrangement of Ts and Fs.  

To the right of the vertical blue separator, we make a column for each symbol.  These are the negation and variable.

Under the variable, we simply transcribe the original values.

Under the negation, we compute the new values.  These values are special because they are the &quot;final&quot; or &quot;main&quot; values, which are the actual value that &lt;math&gt;\neg P&lt;/math&gt; takes, for each given value of ''P''.  

Here is the truth-table for &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{| class=&quot;wikitable&quot;
|+
!''P''  
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;\neg&lt;/math&gt;
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
|-
|T 
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|T
|-
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|T
|F
|}

Be sure to note in which column the main values are recorded.  These are in the second column, under the outer-most &lt;math&gt;\neg&lt;/math&gt;.
{{robelbox|title=Exercise . Triple negation}}
Draw the truth-table for &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise . Double negation equivalence}}
We say that two propositional formulas are &quot;equivalent&quot; if they have the same truth-table.

Because we haven't yet, and cannot yet, state the definition of &quot;propositional formula&quot; then we won't yet try to state a formal definition of &quot;equivalent formulas&quot;.  

However, even at this intuitive and pre-formal stage of explanation, it should be clear that ''P'' and &lt;math&gt;\neg(\neg P)&lt;/math&gt; are equivalent formulas.  This is because the &quot;main&quot; column of the truth-table for each, is 
{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
|-
| T
|-
| F
|}

What formula is equivalent to &lt;math&gt;\neg P&lt;/math&gt;?
{{robelbox/close}}
The following code prints the truth-table using SymPy.  (Recall that you must run the &quot;front-matter&quot; described in the lesson on Python, before running code cells like the one below.)
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    print(str(row[0]) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;

Line 1 generates the truth-table in SymPy.  You don't need to worry about how it does this.

Line 2 prints the header row, using the variable ''x'' instead of ''P'' since SymPy already has ''x'' as a variable.  

Line 3 starts a for-loop to print all of the rows of the table.

Line 4 gives the command to print the current row.  But in SymPy truth tables, each row has two pieces of information.  Therefore we use &lt;code&gt;row[0]&lt;/code&gt; and &lt;code&gt;row[1]&lt;/code&gt; to extract each piece of information.  We place a bar between them in the print command, so that it almost looks like the vertical separator in a table.  

Note that the print-out uses &lt;code&gt;0&lt;/code&gt; and &lt;code&gt;1&lt;/code&gt; instead of truth-values &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;.
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-table of &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Make the print-out of the truth-table a little prettier by writing a function which, given an input list of values 0 or 1, turns it into a corresponding list of &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;. 

Call this function &lt;code&gt;zoToTF&lt;/code&gt;.  

If the function is written correctly, it will cause 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
zoToTF([0,0,1,0,1,1])
&lt;/syntaxhighlight&gt;
to return &lt;code&gt;[&quot;F&quot;,&quot;F&quot;,&quot;T&quot;,&quot;F&quot;,&quot;T&quot;,&quot;T&quot;]&lt;/code&gt;.

Then run the cell 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    tfrow = zoToTF(row[0])
    print(str(tfrow) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;
{{robelbox/close}}
== Conjunction ==

=== Syntax ===

For any two propositional variables, say ''P'' and ''Q'', their conjunction is &lt;math&gt;P\land Q&lt;/math&gt;.  

Again note the importance of parentheses when formulas become complex.  Without parentheses

: &lt;math&gt;\neg P\land Q&lt;/math&gt; 

has two ''equally valid'' readings.  Either 

: &lt;math&gt;\neg(P\land Q)&lt;/math&gt;

or 

: &lt;math&gt;(\neg P)\land Q&lt;/math&gt;

This concern is even more serious than the one that we pointed out for negation.  Here, one cannot reasonably guess the intended formula if it is un-parenthesized.

Moreover, we will see in the section on syntax, that the two formulas have different meaning.  So it is quite important that we distinguish between them.

=== Semantics ===

The semantics of conjunction is that the conjunction should be true when both conjuncts are true, and false otherwise.  This is given in the truth-table below.

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;P&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
|T
|T
|T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|F
|}

Notice the need for four rows now, because there are two variables, each of which could be T or F.  Therefore the full set of possibilities for the pair of variables, ''P'' and ''Q'', are: TT, TF, FT, or FF.  

In general, every time there is an extra variable, the number of rows doubles.  Therefore, if a table requires three variables then its table requires &lt;math&gt;2\cdot 4 = 8&lt;/math&gt; rows.  
{{robelbox|title=Exercise . Three variables}}
Complete the following list of all triples of truth-values for three variables.  

The list is generated by starting from TTT.  

Then take the right-most truth-value and flip it.  If that value was T then we have the subsequent row.

If that value was F then proceed left to the next value and repeat the process.  

TTT

TTF

TFT

TFF

FTT

[row 6: What goes here?]

[row 7: What goes here?]

FFF
{{robelbox/close}}
{{robelbox|title=Exercise . Four variables}}
How many rows are in a truth-table for four variables?

List the first eight rows, using the algorithm of the previous exercise.  
{{robelbox/close}}

As an exercise in reading the truth-table for conjunction, check that you understand:  

When ''P'' is F and ''Q'' is T then &lt;math&gt;P\land Q&lt;/math&gt; is F.  Of course &quot;''P'' is F&quot; is an abbreviation of &quot;''P'' is false&quot;.

We may summarize this by writing &lt;math&gt;F\land T = F&lt;/math&gt;.  Similarly we may summarize the first row of the truth-table by writing &lt;math&gt;T\land T=T&lt;/math&gt;, and so on.  
{{robelbox|title=Exercise . Conjunction commutative}}
Use the previous problem, ''Double negation equivalence'', to prove that &lt;math&gt;P\land Q&lt;/math&gt; is equivalent to &lt;math&gt;Q\land P&lt;/math&gt;.  

(This is fact is called the &quot;commutativity&quot; of conjunction.)
{{robelbox/close}}

----

Here I will produce the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt;, and show you the sequence by which I populate it with values.  

First, draw the basic table and copy the values from the left two columns over to any variables.

''Step 1. Transcribe.''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| |  
| style=&quot;background-color: pink;&quot; | T || 
|style=&quot;background-color: pink;&quot;| T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| |
|style=&quot;background-color: pink;&quot;| T || 
|style=&quot;background-color: pink;&quot;| F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| F
|}

Next, compute the first available operator's column.  In this case, we cannot compute the &lt;math&gt;\land&lt;/math&gt; but we can compute the &lt;math&gt;\neg&lt;/math&gt;, so we start here.

Values are bolded to indicate that they are the values ''from which'' the new values are computed, using the negation rule.

''Step 2. Column 3''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || F
|}

Next, use the values ''under the negation'' and under ''Q'' to compute &lt;math&gt;\land&lt;/math&gt;.  Note again the bolded values to see how the values under &lt;math&gt;\land&lt;/math&gt; were computed.

''Step 3. Column 5''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''T'''
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| T 
| '''T'''
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|}

Note that this last column, column 5, is the &quot;final&quot; value for the proposition.  

----

In the presentation above, we computed entire columns all at once.  This can be nice because it is fast.

However, because this can be confusing, let's just take one row and focus on its computation.  Selecting somewhat randomly, let's focus on the second row.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; | 
! &lt;math&gt;(\neg &lt;/math&gt;
! &lt;math&gt;P)&lt;/math&gt;
! &lt;math&gt;\land&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T 
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| F
| F
|}

To summarize how this row was computed,

1. We transcribe the values for ''P'' and ''Q'' from the left of the blue separator to the right.

2. Because of the parentheses, we first compute the negation.  It reads the T in the fourth column (under ''P'') and results in F in the third column.

3. Next we compute the conjunction, reading F on the left and T on the right.  According to the rule for conjunction, this evaluates to F.
{{robelbox|title=Exercise . Ambiguous without parens}}
Draw the truth-table for &lt;math&gt;\neg(P\land Q)&lt;/math&gt;.

Identify a final value in its truth-table, which does not match one of the final values for the truth-table of &lt;math&gt;(\neg P)\land Q&lt;/math&gt;.  

Conclude that &lt;math&gt;\neg(P\land Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\land Q&lt;/math&gt; have different semantics.  
{{robelbox/close}}
{{robelbox|title=Exercise . Conjunction associativity}}
Show that &lt;math&gt;P\land (Q\land R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land R&lt;/math&gt;.  This fact is called the &quot;associativity&quot; of conjunction.

Use this fact, without computing any more truth-tables, to argue that &lt;math&gt;P\land (Q\land (R\land S))&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land(R\land S)&lt;/math&gt;.  
{{robelbox/close}}

The associativity of conjunction, shown in the exercise above, demonstrates that a sentence does not need parentheses when it is formed only from conjunction.  That is because we might interpret 

: &lt;math&gt;P\land Q\land R&lt;/math&gt; 

as either &lt;math&gt;(P\land Q)\land R&lt;/math&gt; or &lt;math&gt;P\land (Q\land R)&lt;/math&gt;.  But because these are equivalent, then it is not important which one we mean when we write the un-parenthesized version.  

Therefore we will write &lt;math&gt;P\land Q\land R&lt;/math&gt; without additional parentheses.  Likewise we may write &lt;math&gt;P\land Q\land R\land S&lt;/math&gt; without parentheses, because any way of parenthesizing it will always be equivalent to every other way.  

And so on, we may omit parentheses when a sentence is constructed only from conjunction, with any number of variables.  
{{robelbox|title=Exercise|theme=2}}
In SymPy one can generate the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt; with the following cell.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
# Here is an implementation of the zoToTF function described in the previous 
# exercise, which we can use here to make the print-out a little nicer.
def zoToTF(lst):
    out_list = []
    for i in lst:
        if i == 0:
            out_list.append(&quot;F&quot;)
        if i == 1:
            out_list.append(&quot;T&quot;)
    return out_list

ttable = sp.logic.boolalg.truth_table( (~x)&amp;y , [x, y] )
print(&quot;x , y || (~x)&amp;y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;

Now use SymPy to print the truth-table of &lt;math&gt;\neg(P\land(\neg Q))&lt;/math&gt;.
{{robelbox/close}}

== Disjunction ==

=== Syntax ===

The syntax for disjunction is almost identical to that of conjunction.  

For &quot;''P'' or ''Q''&quot; we write &lt;math&gt;P\lor Q&lt;/math&gt;.

=== Semantics ===

The semantics of disjunction, are that the disjunction is true when one or the other disjunct is true.  This is reflected in the truth-table 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\lor&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T
| T
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| T
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| F
| F
|}

Note that a somewhat simpler rule to remember is that &lt;math&gt;P\lor Q&lt;/math&gt; is F when both disjuncts are F.  Otherwise it is T.
{{robelbox|title=Exercise . Disjunction and negation}}
Show that &lt;math&gt;\neg(P\lor Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; are not equivalent.

The proof should very similar to that given for conjunction and negation, earlier.
{{robelbox/close}}
{{robelbox|title=Exercise . All disjunction}}
Decide whether parentheses are needed for a sentence constructed only from the disjunction operator.  

The answer should be decided in a process nearly identical to the one above for conjunction.
{{robelbox/close}}
{{robelbox|title= Exercise . Conjunction and disjunction}}
Investigate whether &lt;math&gt;P\land (Q\lor R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\lor R&lt;/math&gt;.  

Use this to decide whether parentheses are needed for propositions constructed only from conjunction and disjunction.
{{robelbox/close}}
{{robelbox|title=Exercise . De Morgan's}}
The logical law of De Morgan's states that 

: &lt;math&gt;\neg(P\land Q) \equiv (\neg P)\lor (\neg Q)&lt;/math&gt;
and 
: &lt;math&gt;\neg(P\lor Q) \equiv (\neg P)\land (\neg Q)&lt;/math&gt;

This shows an important relationship between these three operators, &lt;math&gt;\neg, \land,\lor&lt;/math&gt;.  

Prove both of these claims of equivalence.  
{{robelbox/close}}
{{robelbox|title= Exercise . De Morgan's with specifics}}
Consider the sentence 
: &quot;2 is not odd and prime.&quot;

''Part 1.''

Is this sentence true?

''Part 2.''

Consider the sentence's abstraction 

: &quot;''t'' is not &lt;math&gt;O(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

Which symbolization seems most correct for the intended meaning of the original sentence?  Either 

: &lt;math&gt;(\neg O(t)) \land P(t)&lt;/math&gt;
or
: &lt;math&gt;\neg(O(t)\land P(t))&lt;/math&gt;
?

Hint: There is a subtle difference between what is communicated by the sentence &quot;2 is not odd and prime&quot; and the sentence &quot;2 is not odd and is prime&quot;.  It is precisely the difference between the two symbolizations.
{{robelbox/close}}

Here is code that prints the truth-table for disjunction in SymPy.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x|y , [x, y] )
print(&quot;x , y || x|y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-tables of the following.
* &lt;math&gt;\neg (P\lor Q)&lt;/math&gt;
* &lt;math&gt;(\neg P)\lor Q&lt;/math&gt;
* &lt;math&gt;(P\land Q)\lor((\neg P)\land (\neg Q))&lt;/math&gt;
{{robelbox/close}}

== Conditional ==

=== Syntax ===

The conditional symbol is &lt;math&gt;\to&lt;/math&gt;, a binary operator. Therefore the syntax for the conditional &quot;if ''P'' then ''Q''&quot; is 

: &lt;math&gt; P\to Q&lt;/math&gt;

=== Semantics ===

The semantics of the conditional are given by the truth-table

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P
! Q 
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\to&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
|| T 
|| T 
|style=&quot;background-color: lightblue;&quot;| 
|| T
| T 
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |  
| T
| F
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|}

A simple way to describe this is: The conditional is true when the antecedent is false, or the consequent is true.

Why would this be the semantics of the conditional?  For negation, conjunction, and disjunction, the reasons for the semantics are probably close to self-evident.  But this rule for the conditional may seem not to obviously match what we think of when we think &quot;If ''P'' then ''Q''.&quot;

The important idea that we mean to capture in the semantics of the conditional, is the &quot;flow of truth&quot; from the antecedent (''P'') to the consequent (''Q'').  

Notice that, in this table, ''P'' is true on the first two rows.  Where ''Q'' is true (row 1), we say that the conditional is true.  This is because, apparently, in a sense the truth of ''P'' successfully flowed to ''Q''.  

On the other hand, where ''P'' is true and ''Q'' is false (row 2), it seems that the flow of truth from ''P'' to ''Q'' was unsuccessful.  Therefore the conditional is marked false here.

But what about the rows at which ''P'' is false (rows 3 and 4)?  Well, since ''P'' is not true then we don't require truth to &quot;flow&quot; toward ''Q'' in this scenario.  In this case, ''Q'' may be true or false, independent of ''P'', and still the conditional should be true.  

Consider the sentence &quot;If a natural number is divisible by 4 then it's divisible by 2.&quot;  This is a true sentence.  

Now consider the instance in which the number is 3.  The sentence, specialized for this instance, would be &quot;If 3 is divisible by 4 then it is divisible by 2.&quot;  This should still be a true sentence, because 3 is not divisible by 4.  In cases where the antecedent is false, we still recognize the conditional as true.  
{{robelbox|title=Exercise . Conditional equivalences}}
''Part 1.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; and also equivalent to &lt;math&gt;(\neg Q)\to (\neg P)&lt;/math&gt;.

''Part 2.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is ''not'' equivalent to &lt;math&gt;Q\to P&lt;/math&gt;.

''Part 3.''

Decide whether &lt;math&gt;P\to (Q\to R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\to Q)\to R&lt;/math&gt;.

''Part 4.''

Decide whether &lt;math&gt;P \to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\to(\neg Q)&lt;/math&gt;.

''Part 5.''

Draw the truth-table for &lt;math&gt;(P\to Q)\land (Q\to P)&lt;/math&gt;.  

Argue that if the formula above is true then ''P'' and ''Q'' must have the same truth-value.  
{{robelbox/close}}

=== An Emphatic Caveat ===

The semantics that we give here for the conditional are, as I've also said at the beginning, the semantics of the conditional ''in mathematical settings''.

It is especially important not to confuse the semantics for this mathematical conditional, with the semantics of every kind of conditional sentence.  The conditional given by the truth-table is called the [https://en.wikipedia.org/wiki/Material_conditional| material conditional].

Some conditional sentences expression causation, like 
: &quot;If you put a magnet next to iron, then there will be an attractive force between them.&quot;

This kind of conditional sentence very much does not have the semantics given by the truth-table above.  

There are also counter-factual conditionals, like 
: &quot;If Franz Ferdinand hadn't been killed Europe wouldn't have gone to war.&quot;

Again, the semantics of such a sentence are not at all like the semantics of the truth-table.  

Later in this course we will discuss causation further, although we will not discuss counter-factuals.  To see a wide-ranging discussion of the semantics of conditionals, one could consult ''A Philosophical Guide to Conditionals'' by Bennett.

== Propositional Formulas ==

Now that we can talk about all of the symbols of propositional logic, we are in a position to define the precise definition of a propositional formula.  By now the intuitive idea should already be clear.  

{{definition|name=propositional formula|value=
We define the set of all propositional formulas, &lt;math&gt;\mathcal F&lt;/math&gt;, in a bottom-up sequence.  This means that we progressively capture more and more formulas with each next set.

First, define &lt;math&gt;\mathcal F_0=\{P_1, P_2, \dots\}&lt;/math&gt; to be any set of propositional variables.  

Next suppose that if ''X'' is any set of expressions, then by writing &lt;math&gt;\neg X&lt;/math&gt; we mean 
: &lt;math&gt;\neg X = \{\neg Q : Q\in X\}&lt;/math&gt;
Likewise define, for any sets of expressions ''X'' and ''Y'', 
: &lt;math&gt;\begin{aligned}
X\land Y &amp;= \{P\land Q:P\in X, Q\in Y\}\\
X\lor Y &amp;= \{P\lor Q: P\in X, Q\in Y\}\\
X\to Y &amp;= \{P\to Q: P\in X, Q\in Y\}
\end{aligned}&lt;/math&gt;

Now if &lt;math&gt;\mathcal F_n&lt;/math&gt; is defined for any natural number ''n'', then we use this to define &lt;math&gt;\mathcal F_{n+1}&lt;/math&gt; as follows.
: &lt;math&gt;\begin{aligned}
\mathcal F_{n+1} = \mathcal F_n&amp;\cup (\neg \mathcal F_n)\\
&amp;\cup (\mathcal F_n\land \mathcal F_n) \\
&amp;\cup (\mathcal F_n\lor \mathcal F_n)\\
&amp;\cup (\mathcal F_n\to\mathcal F_n)
\end{aligned}&lt;/math&gt;

Finally, define &lt;math&gt;\mathcal F = \bigcup_{n=1}^\infty F_n&lt;/math&gt; which is the '''set of all propositional formulas from &lt;math&gt;\mathcal F_0&lt;/math&gt;'''.  

Then any expression, &lt;math&gt;\varphi&lt;/math&gt;, is a '''propositional formula''' if &lt;math&gt;\varphi\in\mathcal F&lt;/math&gt;.
}}
{{robelbox|title=Exercise . Compute a few levels}}
Set &lt;math&gt;\mathcal F_0=\{P, Q\}&lt;/math&gt;.  

Compute &lt;math&gt;\neg \mathcal F_0&lt;/math&gt; and &lt;math&gt;\mathcal F_1&lt;/math&gt;. 

Then repeat the exercise if &lt;math&gt;\mathcal F_0 = \{P\}&lt;/math&gt;.  
{{robelbox/close}}</text>
      <sha1>7pi4i5vurcgtibx1glgyd3uet8hijq1</sha1>
    </revision>
    <revision>
      <id>2624862</id>
      <parentid>2624839</parentid>
      <timestamp>2024-05-02T23:41:22Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Propositional Formulas */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28462" xml:space="preserve">Now that we've seen the basic idea of abstraction and symbolization, we build a formal system around the idea.  

To begin with, because it is simpler, we focus for now on logic at the level of propositions — so-called &quot;propositional logic&quot;.  This means that we do not focus on the quantifiers from the previous lesson, but we do focus on the operators like &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt;.

[[File:Semantic Modeling.png|thumb|A cartoon depiction of a database (right) which holds some data or software objects.  These data are supposed to be in correspondence with things in the real world (left), like buildings or trees or mountains.  This serves as a metaphor for the relationship between syntax and semantics in logic.]]

The most important theme throughout this lesson, is the relationship between ''syntax'' and ''semantics''.  

Syntax is the &quot;dead&quot;, mechanical, &quot;calculational&quot; system of writing some symbols followed (according to a formal rule) by certain other symbols.  To take one example, everything that a computer ''does'' is syntactic.

Semantics is the study of the meaning, which gives those dead symbols life, so to speak.  A computer might display symbols, simply because it was programmed to do so.  But it is the human mind, reading those symbols, which ''gives'' the symbols their meaning.  

== Propositional Variables ==

=== Syntax ===

Because our current interest is in propositions, we will no longer write expressions like ''P(x, y, z)'' as we did in the previous lesson, since we will have nothing to say about the objects ''x, y,'' and ''z''.  We will return to object variables when we get to the lessons on &quot;first-order logic&quot;.   

We will instead use propositional variables, like 

: ''P, Q, R, ...'', 

usually denoted by capital italics Roman letters.  

From a syntactic point of view, these will be the basic &quot;building blocks&quot; or &quot;atoms&quot; which build up to more complex expressions.  

=== Semantics ===

The formal definition of exactly what a propositional variable is, would probably be too technical and cause more confusion than insight at this point.  Therefore let us just rely on the intuition that a propositional variable is an abstraction of any proposition.  

Propositions may be true or false.  

Moreover, they must be precisely one of these two options.  

Therefore the abstraction of a proposition, is essentially the fact that it may be true or false.  

We represent this fact in a so-called &quot;truth table&quot;, like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
| ''P''
|-
| T
|-
| F
|}

The top row shows a propositional variable, and below it, the possible &quot;truth-values&quot; that it could take.  These are true and false, annotated as &quot;T&quot; and &quot;F&quot; respectively.  

At this moment, that might seem like a needlessly elaborate display of a simple idea.  However, things are about to get complicated soon, and it will be valuable to have tables to organize our information.

=== Caveats ===

Note that the semantics we are giving here are the semantics of mathematics, but they are not exactly the same as the semantics of everyday speech.  

For one thing, the semantics of everyday speech includes much more than propositions.  

But also, there are some philosophical disputes about exactly what the semantics of statements are.  One could make nonsensical statements, like &quot;green is round&quot;, which seems not just false but meaningless.  In general, [https://en.wikipedia.org/wiki/Category_mistake| category mistakes] of all kinds often seem neither true nor false but meaningless.

Another example is &quot;the present king of France is bald.&quot;  There is (as of writing this) no present kind of France.  Some philosophers argue that this sentence is technically false (see [https://en.wikipedia.org/wiki/Theory_of_descriptions| the theory of descriptions]), but many philosophers and linguists reject this analysis.  However, there is no consensus on how to regard such a sentence.  

One could go on, but let's not get distracted.  Just be aware that the semantics presented here are those used in mathematical settings, and that is a very useful setting for analyzing a wide range of real-world phenomena.  But these semantics are narrowly limited.

== Negation ==

=== Syntax ===

The negation of a propositional variable, like ''P'', is written as &lt;math&gt;\neg P&lt;/math&gt;.  

Notice that although ''P'' is a propositional variable, &lt;math&gt;\neg P&lt;/math&gt; is not.  Rather, it is an example of what we will eventually call a ''propositional formula''.  We are not yet in a position to define this phrase exactly; but toward the end of this lesson, we will.

A propositional formula is essentially just anything that is built from propositional variables using propositional operators (negation, conjunction, disjunction, and conditional).  

Moreover, notice that we may negate ''any'' propositional formula, not just variables.   Therefore 

: &lt;math&gt;\neg (\neg P)&lt;/math&gt;
and 
: &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;

are propositional formulas too.  

Note the use of parentheses, which will become increasingly important as the complexity of propositional formulas grows.  We may write &lt;math&gt;(\neg P)&lt;/math&gt; if we wanted to, or &lt;math&gt;(\neg(\neg P))&lt;/math&gt;.  But we also allow ourselves to omit the outer-most parentheses.  

=== Semantics ===

The semantics of negation is to switch true to false, and false to true.  

Therefore we write the truth table of negation like so 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! &lt;math&gt;\neg P&lt;/math&gt;
|-
| F
|-
| T
|}

One often wants to juxtapose this, with the truth-table from which the negated values were derived, to help see what happened here.  

Therefore we prefer to write the truth-table instead like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot; 
|-
! |''P'' 
!style=&quot;background-color: lightblue;&quot; |
! |&lt;math&gt;\neg &lt;/math&gt; 
! |&lt;math&gt;P&lt;/math&gt;
|-
| T|| style=&quot;background-color: lightblue;&quot; |
| F|| T
|-
| F|| style=&quot;background-color: lightblue;&quot;|
|T||F
|}

The left-most column simply records the original arrangement of Ts and Fs.  

To the right of the vertical blue separator, we make a column for each symbol.  These are the negation and variable.

Under the variable, we simply transcribe the original values.

Under the negation, we compute the new values.  These values are special because they are the &quot;final&quot; or &quot;main&quot; values, which are the actual value that &lt;math&gt;\neg P&lt;/math&gt; takes, for each given value of ''P''.  

Here is the truth-table for &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{| class=&quot;wikitable&quot;
|+
!''P''  
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;\neg&lt;/math&gt;
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
|-
|T 
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|T
|-
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|T
|F
|}

Be sure to note in which column the main values are recorded.  These are in the second column, under the outer-most &lt;math&gt;\neg&lt;/math&gt;.
{{robelbox|title=Exercise . Triple negation}}
Draw the truth-table for &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise . Double negation equivalence}}
We say that two propositional formulas are &quot;equivalent&quot; if they have the same truth-table.

Because we haven't yet, and cannot yet, state the definition of &quot;propositional formula&quot; then we won't yet try to state a formal definition of &quot;equivalent formulas&quot;.  

However, even at this intuitive and pre-formal stage of explanation, it should be clear that ''P'' and &lt;math&gt;\neg(\neg P)&lt;/math&gt; are equivalent formulas.  This is because the &quot;main&quot; column of the truth-table for each, is 
{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
|-
| T
|-
| F
|}

What formula is equivalent to &lt;math&gt;\neg P&lt;/math&gt;?
{{robelbox/close}}
The following code prints the truth-table using SymPy.  (Recall that you must run the &quot;front-matter&quot; described in the lesson on Python, before running code cells like the one below.)
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    print(str(row[0]) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;

Line 1 generates the truth-table in SymPy.  You don't need to worry about how it does this.

Line 2 prints the header row, using the variable ''x'' instead of ''P'' since SymPy already has ''x'' as a variable.  

Line 3 starts a for-loop to print all of the rows of the table.

Line 4 gives the command to print the current row.  But in SymPy truth tables, each row has two pieces of information.  Therefore we use &lt;code&gt;row[0]&lt;/code&gt; and &lt;code&gt;row[1]&lt;/code&gt; to extract each piece of information.  We place a bar between them in the print command, so that it almost looks like the vertical separator in a table.  

Note that the print-out uses &lt;code&gt;0&lt;/code&gt; and &lt;code&gt;1&lt;/code&gt; instead of truth-values &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;.
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-table of &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Make the print-out of the truth-table a little prettier by writing a function which, given an input list of values 0 or 1, turns it into a corresponding list of &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;. 

Call this function &lt;code&gt;zoToTF&lt;/code&gt;.  

If the function is written correctly, it will cause 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
zoToTF([0,0,1,0,1,1])
&lt;/syntaxhighlight&gt;
to return &lt;code&gt;[&quot;F&quot;,&quot;F&quot;,&quot;T&quot;,&quot;F&quot;,&quot;T&quot;,&quot;T&quot;]&lt;/code&gt;.

Then run the cell 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    tfrow = zoToTF(row[0])
    print(str(tfrow) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;
{{robelbox/close}}
== Conjunction ==

=== Syntax ===

For any two propositional variables, say ''P'' and ''Q'', their conjunction is &lt;math&gt;P\land Q&lt;/math&gt;.  

Again note the importance of parentheses when formulas become complex.  Without parentheses

: &lt;math&gt;\neg P\land Q&lt;/math&gt; 

has two ''equally valid'' readings.  Either 

: &lt;math&gt;\neg(P\land Q)&lt;/math&gt;

or 

: &lt;math&gt;(\neg P)\land Q&lt;/math&gt;

This concern is even more serious than the one that we pointed out for negation.  Here, one cannot reasonably guess the intended formula if it is un-parenthesized.

Moreover, we will see in the section on syntax, that the two formulas have different meaning.  So it is quite important that we distinguish between them.

=== Semantics ===

The semantics of conjunction is that the conjunction should be true when both conjuncts are true, and false otherwise.  This is given in the truth-table below.

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;P&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
|T
|T
|T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|F
|}

Notice the need for four rows now, because there are two variables, each of which could be T or F.  Therefore the full set of possibilities for the pair of variables, ''P'' and ''Q'', are: TT, TF, FT, or FF.  

In general, every time there is an extra variable, the number of rows doubles.  Therefore, if a table requires three variables then its table requires &lt;math&gt;2\cdot 4 = 8&lt;/math&gt; rows.  
{{robelbox|title=Exercise . Three variables}}
Complete the following list of all triples of truth-values for three variables.  

The list is generated by starting from TTT.  

Then take the right-most truth-value and flip it.  If that value was T then we have the subsequent row.

If that value was F then proceed left to the next value and repeat the process.  

TTT

TTF

TFT

TFF

FTT

[row 6: What goes here?]

[row 7: What goes here?]

FFF
{{robelbox/close}}
{{robelbox|title=Exercise . Four variables}}
How many rows are in a truth-table for four variables?

List the first eight rows, using the algorithm of the previous exercise.  
{{robelbox/close}}

As an exercise in reading the truth-table for conjunction, check that you understand:  

When ''P'' is F and ''Q'' is T then &lt;math&gt;P\land Q&lt;/math&gt; is F.  Of course &quot;''P'' is F&quot; is an abbreviation of &quot;''P'' is false&quot;.

We may summarize this by writing &lt;math&gt;F\land T = F&lt;/math&gt;.  Similarly we may summarize the first row of the truth-table by writing &lt;math&gt;T\land T=T&lt;/math&gt;, and so on.  
{{robelbox|title=Exercise . Conjunction commutative}}
Use the previous problem, ''Double negation equivalence'', to prove that &lt;math&gt;P\land Q&lt;/math&gt; is equivalent to &lt;math&gt;Q\land P&lt;/math&gt;.  

(This is fact is called the &quot;commutativity&quot; of conjunction.)
{{robelbox/close}}

----

Here I will produce the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt;, and show you the sequence by which I populate it with values.  

First, draw the basic table and copy the values from the left two columns over to any variables.

''Step 1. Transcribe.''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| |  
| style=&quot;background-color: pink;&quot; | T || 
|style=&quot;background-color: pink;&quot;| T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| |
|style=&quot;background-color: pink;&quot;| T || 
|style=&quot;background-color: pink;&quot;| F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| F
|}

Next, compute the first available operator's column.  In this case, we cannot compute the &lt;math&gt;\land&lt;/math&gt; but we can compute the &lt;math&gt;\neg&lt;/math&gt;, so we start here.

Values are bolded to indicate that they are the values ''from which'' the new values are computed, using the negation rule.

''Step 2. Column 3''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || F
|}

Next, use the values ''under the negation'' and under ''Q'' to compute &lt;math&gt;\land&lt;/math&gt;.  Note again the bolded values to see how the values under &lt;math&gt;\land&lt;/math&gt; were computed.

''Step 3. Column 5''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''T'''
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| T 
| '''T'''
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|}

Note that this last column, column 5, is the &quot;final&quot; value for the proposition.  

----

In the presentation above, we computed entire columns all at once.  This can be nice because it is fast.

However, because this can be confusing, let's just take one row and focus on its computation.  Selecting somewhat randomly, let's focus on the second row.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; | 
! &lt;math&gt;(\neg &lt;/math&gt;
! &lt;math&gt;P)&lt;/math&gt;
! &lt;math&gt;\land&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T 
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| F
| F
|}

To summarize how this row was computed,

1. We transcribe the values for ''P'' and ''Q'' from the left of the blue separator to the right.

2. Because of the parentheses, we first compute the negation.  It reads the T in the fourth column (under ''P'') and results in F in the third column.

3. Next we compute the conjunction, reading F on the left and T on the right.  According to the rule for conjunction, this evaluates to F.
{{robelbox|title=Exercise . Ambiguous without parens}}
Draw the truth-table for &lt;math&gt;\neg(P\land Q)&lt;/math&gt;.

Identify a final value in its truth-table, which does not match one of the final values for the truth-table of &lt;math&gt;(\neg P)\land Q&lt;/math&gt;.  

Conclude that &lt;math&gt;\neg(P\land Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\land Q&lt;/math&gt; have different semantics.  
{{robelbox/close}}
{{robelbox|title=Exercise . Conjunction associativity}}
Show that &lt;math&gt;P\land (Q\land R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land R&lt;/math&gt;.  This fact is called the &quot;associativity&quot; of conjunction.

Use this fact, without computing any more truth-tables, to argue that &lt;math&gt;P\land (Q\land (R\land S))&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land(R\land S)&lt;/math&gt;.  
{{robelbox/close}}

The associativity of conjunction, shown in the exercise above, demonstrates that a sentence does not need parentheses when it is formed only from conjunction.  That is because we might interpret 

: &lt;math&gt;P\land Q\land R&lt;/math&gt; 

as either &lt;math&gt;(P\land Q)\land R&lt;/math&gt; or &lt;math&gt;P\land (Q\land R)&lt;/math&gt;.  But because these are equivalent, then it is not important which one we mean when we write the un-parenthesized version.  

Therefore we will write &lt;math&gt;P\land Q\land R&lt;/math&gt; without additional parentheses.  Likewise we may write &lt;math&gt;P\land Q\land R\land S&lt;/math&gt; without parentheses, because any way of parenthesizing it will always be equivalent to every other way.  

And so on, we may omit parentheses when a sentence is constructed only from conjunction, with any number of variables.  
{{robelbox|title=Exercise|theme=2}}
In SymPy one can generate the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt; with the following cell.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
# Here is an implementation of the zoToTF function described in the previous 
# exercise, which we can use here to make the print-out a little nicer.
def zoToTF(lst):
    out_list = []
    for i in lst:
        if i == 0:
            out_list.append(&quot;F&quot;)
        if i == 1:
            out_list.append(&quot;T&quot;)
    return out_list

ttable = sp.logic.boolalg.truth_table( (~x)&amp;y , [x, y] )
print(&quot;x , y || (~x)&amp;y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;

Now use SymPy to print the truth-table of &lt;math&gt;\neg(P\land(\neg Q))&lt;/math&gt;.
{{robelbox/close}}

== Disjunction ==

=== Syntax ===

The syntax for disjunction is almost identical to that of conjunction.  

For &quot;''P'' or ''Q''&quot; we write &lt;math&gt;P\lor Q&lt;/math&gt;.

=== Semantics ===

The semantics of disjunction, are that the disjunction is true when one or the other disjunct is true.  This is reflected in the truth-table 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\lor&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T
| T
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| T
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| F
| F
|}

Note that a somewhat simpler rule to remember is that &lt;math&gt;P\lor Q&lt;/math&gt; is F when both disjuncts are F.  Otherwise it is T.
{{robelbox|title=Exercise . Disjunction and negation}}
Show that &lt;math&gt;\neg(P\lor Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; are not equivalent.

The proof should very similar to that given for conjunction and negation, earlier.
{{robelbox/close}}
{{robelbox|title=Exercise . All disjunction}}
Decide whether parentheses are needed for a sentence constructed only from the disjunction operator.  

The answer should be decided in a process nearly identical to the one above for conjunction.
{{robelbox/close}}
{{robelbox|title= Exercise . Conjunction and disjunction}}
Investigate whether &lt;math&gt;P\land (Q\lor R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\lor R&lt;/math&gt;.  

Use this to decide whether parentheses are needed for propositions constructed only from conjunction and disjunction.
{{robelbox/close}}
{{robelbox|title=Exercise . De Morgan's}}
The logical law of De Morgan's states that 

: &lt;math&gt;\neg(P\land Q) \equiv (\neg P)\lor (\neg Q)&lt;/math&gt;
and 
: &lt;math&gt;\neg(P\lor Q) \equiv (\neg P)\land (\neg Q)&lt;/math&gt;

This shows an important relationship between these three operators, &lt;math&gt;\neg, \land,\lor&lt;/math&gt;.  

Prove both of these claims of equivalence.  
{{robelbox/close}}
{{robelbox|title= Exercise . De Morgan's with specifics}}
Consider the sentence 
: &quot;2 is not odd and prime.&quot;

''Part 1.''

Is this sentence true?

''Part 2.''

Consider the sentence's abstraction 

: &quot;''t'' is not &lt;math&gt;O(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

Which symbolization seems most correct for the intended meaning of the original sentence?  Either 

: &lt;math&gt;(\neg O(t)) \land P(t)&lt;/math&gt;
or
: &lt;math&gt;\neg(O(t)\land P(t))&lt;/math&gt;
?

Hint: There is a subtle difference between what is communicated by the sentence &quot;2 is not odd and prime&quot; and the sentence &quot;2 is not odd and is prime&quot;.  It is precisely the difference between the two symbolizations.
{{robelbox/close}}

Here is code that prints the truth-table for disjunction in SymPy.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x|y , [x, y] )
print(&quot;x , y || x|y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-tables of the following.
* &lt;math&gt;\neg (P\lor Q)&lt;/math&gt;
* &lt;math&gt;(\neg P)\lor Q&lt;/math&gt;
* &lt;math&gt;(P\land Q)\lor((\neg P)\land (\neg Q))&lt;/math&gt;
{{robelbox/close}}

== Conditional ==

=== Syntax ===

The conditional symbol is &lt;math&gt;\to&lt;/math&gt;, a binary operator. Therefore the syntax for the conditional &quot;if ''P'' then ''Q''&quot; is 

: &lt;math&gt; P\to Q&lt;/math&gt;

=== Semantics ===

The semantics of the conditional are given by the truth-table

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P
! Q 
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\to&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
|| T 
|| T 
|style=&quot;background-color: lightblue;&quot;| 
|| T
| T 
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |  
| T
| F
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|}

A simple way to describe this is: The conditional is true when the antecedent is false, or the consequent is true.

Why would this be the semantics of the conditional?  For negation, conjunction, and disjunction, the reasons for the semantics are probably close to self-evident.  But this rule for the conditional may seem not to obviously match what we think of when we think &quot;If ''P'' then ''Q''.&quot;

The important idea that we mean to capture in the semantics of the conditional, is the &quot;flow of truth&quot; from the antecedent (''P'') to the consequent (''Q'').  

Notice that, in this table, ''P'' is true on the first two rows.  Where ''Q'' is true (row 1), we say that the conditional is true.  This is because, apparently, in a sense the truth of ''P'' successfully flowed to ''Q''.  

On the other hand, where ''P'' is true and ''Q'' is false (row 2), it seems that the flow of truth from ''P'' to ''Q'' was unsuccessful.  Therefore the conditional is marked false here.

But what about the rows at which ''P'' is false (rows 3 and 4)?  Well, since ''P'' is not true then we don't require truth to &quot;flow&quot; toward ''Q'' in this scenario.  In this case, ''Q'' may be true or false, independent of ''P'', and still the conditional should be true.  

Consider the sentence &quot;If a natural number is divisible by 4 then it's divisible by 2.&quot;  This is a true sentence.  

Now consider the instance in which the number is 3.  The sentence, specialized for this instance, would be &quot;If 3 is divisible by 4 then it is divisible by 2.&quot;  This should still be a true sentence, because 3 is not divisible by 4.  In cases where the antecedent is false, we still recognize the conditional as true.  
{{robelbox|title=Exercise . Conditional equivalences}}
''Part 1.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; and also equivalent to &lt;math&gt;(\neg Q)\to (\neg P)&lt;/math&gt;.

''Part 2.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is ''not'' equivalent to &lt;math&gt;Q\to P&lt;/math&gt;.

''Part 3.''

Decide whether &lt;math&gt;P\to (Q\to R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\to Q)\to R&lt;/math&gt;.

''Part 4.''

Decide whether &lt;math&gt;P \to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\to(\neg Q)&lt;/math&gt;.

''Part 5.''

Draw the truth-table for &lt;math&gt;(P\to Q)\land (Q\to P)&lt;/math&gt;.  

Argue that if the formula above is true then ''P'' and ''Q'' must have the same truth-value.  
{{robelbox/close}}

=== An Emphatic Caveat ===

The semantics that we give here for the conditional are, as I've also said at the beginning, the semantics of the conditional ''in mathematical settings''.

It is especially important not to confuse the semantics for this mathematical conditional, with the semantics of every kind of conditional sentence.  The conditional given by the truth-table is called the [https://en.wikipedia.org/wiki/Material_conditional| material conditional].

Some conditional sentences expression causation, like 
: &quot;If you put a magnet next to iron, then there will be an attractive force between them.&quot;

This kind of conditional sentence very much does not have the semantics given by the truth-table above.  

There are also counter-factual conditionals, like 
: &quot;If Franz Ferdinand hadn't been killed Europe wouldn't have gone to war.&quot;

Again, the semantics of such a sentence are not at all like the semantics of the truth-table.  

Later in this course we will discuss causation further, although we will not discuss counter-factuals.  To see a wide-ranging discussion of the semantics of conditionals, one could consult ''A Philosophical Guide to Conditionals'' by Bennett.

== Propositional Formulas ==

Now that we can talk about all of the symbols of propositional logic, we are in a position to define the precise definition of a propositional formula.  

The following definition is quite technical, and for some students this may be hard to process.  However, the intuitive idea of what a propositional formula is, should already be clear.  

The following are several propositional formulas.  They are just the kinds of things we made truth-tables for, earlier.

: &lt;math&gt; (P\to (\neg Q))&lt;/math&gt;
: &lt;math&gt; ((P\land Q)\land (P\lor Q))&lt;/math&gt;

Now these formulas have extra surrounding parentheses, unlike how we wrote them earlier.  These parentheses are unnecessary for us to read and understand the formula, but having them makes it easier to state a formal definition as we do below.  

{{definition|name=propositional formula|value=
We define the set of all propositional formulas, &lt;math&gt;\mathcal F&lt;/math&gt;, in a bottom-up sequence.  This means that we progressively capture more and more formulas with each next set.

First, define &lt;math&gt;\mathcal F_0=\{P_1, P_2, \dots\}&lt;/math&gt; to be any set of propositional variables.  

Next suppose that if ''X'' is any set of expressions, then by writing &lt;math&gt;\neg X&lt;/math&gt; we mean 
: &lt;math&gt;(\neg X) = \{(\neg Q) : Q\in X\}&lt;/math&gt;
Likewise define, for any sets of expressions ''X'' and ''Y'', 
: &lt;math&gt;\begin{aligned}
(X\land Y) &amp;= \{(P\land Q):P\in X, Q\in Y\}\\
(X\lor Y) &amp;= \{(P\lor Q): P\in X, Q\in Y\}\\
(X\to Y) &amp;= \{(P\to Q): P\in X, Q\in Y\}
\end{aligned}&lt;/math&gt;

Now if &lt;math&gt;\mathcal F_n&lt;/math&gt; is defined for any natural number ''n'', then we use this to define &lt;math&gt;\mathcal F_{n+1}&lt;/math&gt; as follows.
: &lt;math&gt;\begin{aligned}
\mathcal F_{n+1} = \mathcal F_n&amp;\cup (\neg \mathcal F_n)\\
&amp;\cup (\mathcal F_n\land \mathcal F_n) \\
&amp;\cup (\mathcal F_n\lor \mathcal F_n)\\
&amp;\cup (\mathcal F_n\to\mathcal F_n)
\end{aligned}&lt;/math&gt;

Finally, define &lt;math&gt;\mathcal F = \bigcup_{n=0}^\infty F_n&lt;/math&gt; which is the '''set of all propositional formulas from &lt;math&gt;\mathcal F_0&lt;/math&gt;'''.  

Then any expression, &lt;math&gt;\varphi&lt;/math&gt;, is a '''propositional formula''' if &lt;math&gt;\varphi\in\mathcal F&lt;/math&gt;.
}}
{{robelbox|title=Exercise . Compute a few levels}}
Set &lt;math&gt;\mathcal F_0=\{P, Q\}&lt;/math&gt;.  

Compute &lt;math&gt;\neg \mathcal F_0&lt;/math&gt; and &lt;math&gt;\mathcal F_1&lt;/math&gt;. 

Then repeat the exercise if &lt;math&gt;\mathcal F_0 = \{P\}&lt;/math&gt;.  
{{robelbox/close}}</text>
      <sha1>ks589ts20vdf5l3en0l2chk8jfac7zv</sha1>
    </revision>
    <revision>
      <id>2624889</id>
      <parentid>2624862</parentid>
      <timestamp>2024-05-03T02:23:14Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Propositional Formulas */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="32379" xml:space="preserve">Now that we've seen the basic idea of abstraction and symbolization, we build a formal system around the idea.  

To begin with, because it is simpler, we focus for now on logic at the level of propositions — so-called &quot;propositional logic&quot;.  This means that we do not focus on the quantifiers from the previous lesson, but we do focus on the operators like &lt;math&gt;\land,\lor,\neg,\to&lt;/math&gt;.

[[File:Semantic Modeling.png|thumb|A cartoon depiction of a database (right) which holds some data or software objects.  These data are supposed to be in correspondence with things in the real world (left), like buildings or trees or mountains.  This serves as a metaphor for the relationship between syntax and semantics in logic.]]

The most important theme throughout this lesson, is the relationship between ''syntax'' and ''semantics''.  

Syntax is the &quot;dead&quot;, mechanical, &quot;calculational&quot; system of writing some symbols followed (according to a formal rule) by certain other symbols.  To take one example, everything that a computer ''does'' is syntactic.

Semantics is the study of the meaning, which gives those dead symbols life, so to speak.  A computer might display symbols, simply because it was programmed to do so.  But it is the human mind, reading those symbols, which ''gives'' the symbols their meaning.  

== Propositional Variables ==

=== Syntax ===

Because our current interest is in propositions, we will no longer write expressions like ''P(x, y, z)'' as we did in the previous lesson, since we will have nothing to say about the objects ''x, y,'' and ''z''.  We will return to object variables when we get to the lessons on &quot;first-order logic&quot;.   

We will instead use propositional variables, like 

: ''P, Q, R, ...'', 

usually denoted by capital italics Roman letters.  

From a syntactic point of view, these will be the basic &quot;building blocks&quot; or &quot;atoms&quot; which build up to more complex expressions.  

=== Semantics ===

The formal definition of exactly what a propositional variable is, would probably be too technical and cause more confusion than insight at this point.  Therefore let us just rely on the intuition that a propositional variable is an abstraction of any proposition.  

Propositions may be true or false.  

Moreover, they must be precisely one of these two options.  

Therefore the abstraction of a proposition, is essentially the fact that it may be true or false.  

We represent this fact in a so-called &quot;truth table&quot;, like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
| ''P''
|-
| T
|-
| F
|}

The top row shows a propositional variable, and below it, the possible &quot;truth-values&quot; that it could take.  These are true and false, annotated as &quot;T&quot; and &quot;F&quot; respectively.  

At this moment, that might seem like a needlessly elaborate display of a simple idea.  However, things are about to get complicated soon, and it will be valuable to have tables to organize our information.

=== Caveats ===

Note that the semantics we are giving here are the semantics of mathematics, but they are not exactly the same as the semantics of everyday speech.  

For one thing, the semantics of everyday speech includes much more than propositions.  

But also, there are some philosophical disputes about exactly what the semantics of statements are.  One could make nonsensical statements, like &quot;green is round&quot;, which seems not just false but meaningless.  In general, [https://en.wikipedia.org/wiki/Category_mistake| category mistakes] of all kinds often seem neither true nor false but meaningless.

Another example is &quot;the present king of France is bald.&quot;  There is (as of writing this) no present kind of France.  Some philosophers argue that this sentence is technically false (see [https://en.wikipedia.org/wiki/Theory_of_descriptions| the theory of descriptions]), but many philosophers and linguists reject this analysis.  However, there is no consensus on how to regard such a sentence.  

One could go on, but let's not get distracted.  Just be aware that the semantics presented here are those used in mathematical settings, and that is a very useful setting for analyzing a wide range of real-world phenomena.  But these semantics are narrowly limited.

== Negation ==

=== Syntax ===

The negation of a propositional variable, like ''P'', is written as &lt;math&gt;\neg P&lt;/math&gt;.  

Notice that although ''P'' is a propositional variable, &lt;math&gt;\neg P&lt;/math&gt; is not.  Rather, it is an example of what we will eventually call a ''propositional formula''.  We are not yet in a position to define this phrase exactly; but toward the end of this lesson, we will.

A propositional formula is essentially just anything that is built from propositional variables using propositional operators (negation, conjunction, disjunction, and conditional).  

Moreover, notice that we may negate ''any'' propositional formula, not just variables.   Therefore 

: &lt;math&gt;\neg (\neg P)&lt;/math&gt;
and 
: &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;

are propositional formulas too.  

Note the use of parentheses, which will become increasingly important as the complexity of propositional formulas grows.  We may write &lt;math&gt;(\neg P)&lt;/math&gt; if we wanted to, or &lt;math&gt;(\neg(\neg P))&lt;/math&gt;.  But we also allow ourselves to omit the outer-most parentheses.  

=== Semantics ===

The semantics of negation is to switch true to false, and false to true.  

Therefore we write the truth table of negation like so 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! &lt;math&gt;\neg P&lt;/math&gt;
|-
| F
|-
| T
|}

One often wants to juxtapose this, with the truth-table from which the negated values were derived, to help see what happened here.  

Therefore we prefer to write the truth-table instead like so.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot; 
|-
! |''P'' 
!style=&quot;background-color: lightblue;&quot; |
! |&lt;math&gt;\neg &lt;/math&gt; 
! |&lt;math&gt;P&lt;/math&gt;
|-
| T|| style=&quot;background-color: lightblue;&quot; |
| F|| T
|-
| F|| style=&quot;background-color: lightblue;&quot;|
|T||F
|}

The left-most column simply records the original arrangement of Ts and Fs.  

To the right of the vertical blue separator, we make a column for each symbol.  These are the negation and variable.

Under the variable, we simply transcribe the original values.

Under the negation, we compute the new values.  These values are special because they are the &quot;final&quot; or &quot;main&quot; values, which are the actual value that &lt;math&gt;\neg P&lt;/math&gt; takes, for each given value of ''P''.  

Here is the truth-table for &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{| class=&quot;wikitable&quot;
|+
!''P''  
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;\neg&lt;/math&gt;
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
|-
|T 
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|T
|-
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|T
|F
|}

Be sure to note in which column the main values are recorded.  These are in the second column, under the outer-most &lt;math&gt;\neg&lt;/math&gt;.
{{robelbox|title=Exercise . Triple negation}}
Draw the truth-table for &lt;math&gt;\neg(\neg(\neg P))&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise . Double negation equivalence}}
We say that two propositional formulas are &quot;equivalent&quot; if they have the same truth-table.

Because we haven't yet, and cannot yet, state the definition of &quot;propositional formula&quot; then we won't yet try to state a formal definition of &quot;equivalent formulas&quot;.  

However, even at this intuitive and pre-formal stage of explanation, it should be clear that ''P'' and &lt;math&gt;\neg(\neg P)&lt;/math&gt; are equivalent formulas.  This is because the &quot;main&quot; column of the truth-table for each, is 
{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
|-
| T
|-
| F
|}

What formula is equivalent to &lt;math&gt;\neg P&lt;/math&gt;?
{{robelbox/close}}
The following code prints the truth-table using SymPy.  (Recall that you must run the &quot;front-matter&quot; described in the lesson on Python, before running code cells like the one below.)
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    print(str(row[0]) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;

Line 1 generates the truth-table in SymPy.  You don't need to worry about how it does this.

Line 2 prints the header row, using the variable ''x'' instead of ''P'' since SymPy already has ''x'' as a variable.  

Line 3 starts a for-loop to print all of the rows of the table.

Line 4 gives the command to print the current row.  But in SymPy truth tables, each row has two pieces of information.  Therefore we use &lt;code&gt;row[0]&lt;/code&gt; and &lt;code&gt;row[1]&lt;/code&gt; to extract each piece of information.  We place a bar between them in the print command, so that it almost looks like the vertical separator in a table.  

Note that the print-out uses &lt;code&gt;0&lt;/code&gt; and &lt;code&gt;1&lt;/code&gt; instead of truth-values &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;.
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-table of &lt;math&gt;\neg(\neg P)&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Make the print-out of the truth-table a little prettier by writing a function which, given an input list of values 0 or 1, turns it into a corresponding list of &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;T&lt;/code&gt;. 

Call this function &lt;code&gt;zoToTF&lt;/code&gt;.  

If the function is written correctly, it will cause 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
zoToTF([0,0,1,0,1,1])
&lt;/syntaxhighlight&gt;
to return &lt;code&gt;[&quot;F&quot;,&quot;F&quot;,&quot;T&quot;,&quot;F&quot;,&quot;T&quot;,&quot;T&quot;]&lt;/code&gt;.

Then run the cell 
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x, [x] )
print(&quot;x | ~x&quot;)
for row in ttable:
    tfrow = zoToTF(row[0])
    print(str(tfrow) + &quot; | &quot; + str(row[1]))
&lt;/syntaxhighlight&gt;
{{robelbox/close}}
== Conjunction ==

=== Syntax ===

For any two propositional variables, say ''P'' and ''Q'', their conjunction is &lt;math&gt;P\land Q&lt;/math&gt;.  

Again note the importance of parentheses when formulas become complex.  Without parentheses

: &lt;math&gt;\neg P\land Q&lt;/math&gt; 

has two ''equally valid'' readings.  Either 

: &lt;math&gt;\neg(P\land Q)&lt;/math&gt;

or 

: &lt;math&gt;(\neg P)\land Q&lt;/math&gt;

This concern is even more serious than the one that we pointed out for negation.  Here, one cannot reasonably guess the intended formula if it is un-parenthesized.

Moreover, we will see in the section on syntax, that the two formulas have different meaning.  So it is quite important that we distinguish between them.

=== Semantics ===

The semantics of conjunction is that the conjunction should be true when both conjuncts are true, and false otherwise.  This is given in the truth-table below.

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;P&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
|T
|T
|T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
|T
|F
|F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
|F
|F
|F
|}

Notice the need for four rows now, because there are two variables, each of which could be T or F.  Therefore the full set of possibilities for the pair of variables, ''P'' and ''Q'', are: TT, TF, FT, or FF.  

In general, every time there is an extra variable, the number of rows doubles.  Therefore, if a table requires three variables then its table requires &lt;math&gt;2\cdot 4 = 8&lt;/math&gt; rows.  
{{robelbox|title=Exercise . Three variables}}
Complete the following list of all triples of truth-values for three variables.  

The list is generated by starting from TTT.  

Then take the right-most truth-value and flip it.  If that value was T then we have the subsequent row.

If that value was F then proceed left to the next value and repeat the process.  

TTT

TTF

TFT

TFF

FTT

[row 6: What goes here?]

[row 7: What goes here?]

FFF
{{robelbox/close}}
{{robelbox|title=Exercise . Four variables}}
How many rows are in a truth-table for four variables?

List the first eight rows, using the algorithm of the previous exercise.  
{{robelbox/close}}

As an exercise in reading the truth-table for conjunction, check that you understand:  

When ''P'' is F and ''Q'' is T then &lt;math&gt;P\land Q&lt;/math&gt; is F.  Of course &quot;''P'' is F&quot; is an abbreviation of &quot;''P'' is false&quot;.

We may summarize this by writing &lt;math&gt;F\land T = F&lt;/math&gt;.  Similarly we may summarize the first row of the truth-table by writing &lt;math&gt;T\land T=T&lt;/math&gt;, and so on.  
{{robelbox|title=Exercise . Conjunction commutative}}
Use the previous problem, ''Double negation equivalence'', to prove that &lt;math&gt;P\land Q&lt;/math&gt; is equivalent to &lt;math&gt;Q\land P&lt;/math&gt;.  

(This is fact is called the &quot;commutativity&quot; of conjunction.)
{{robelbox/close}}

----

Here I will produce the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt;, and show you the sequence by which I populate it with values.  

First, draw the basic table and copy the values from the left two columns over to any variables.

''Step 1. Transcribe.''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| |  
| style=&quot;background-color: pink;&quot; | T || 
|style=&quot;background-color: pink;&quot;| T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| |
|style=&quot;background-color: pink;&quot;| T || 
|style=&quot;background-color: pink;&quot;| F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| | 
|style=&quot;background-color: pink;&quot;| F || 
|style=&quot;background-color: pink;&quot;| F
|}

Next, compute the first available operator's column.  In this case, we cannot compute the &lt;math&gt;\land&lt;/math&gt; but we can compute the &lt;math&gt;\neg&lt;/math&gt;, so we start here.

Values are bolded to indicate that they are the values ''from which'' the new values are computed, using the negation rule.

''Step 2. Column 3''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || T
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| F 
| '''T''' || || F
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || T
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| style=&quot;background-color: pink;&quot;| T 
| '''F''' || || F
|}

Next, use the values ''under the negation'' and under ''Q'' to compute &lt;math&gt;\land&lt;/math&gt;.  Note again the bolded values to see how the values under &lt;math&gt;\land&lt;/math&gt; were computed.

''Step 3. Column 5''

{| class=&quot;wikitable&quot;
|+
!''P''  
!''Q''
!style=&quot;background-color: lightblue;&quot;|
!&lt;math&gt;(\neg&lt;/math&gt;
!&lt;math&gt;P)&lt;/math&gt;
!&lt;math&gt;\land&lt;/math&gt;
!&lt;math&gt;Q&lt;/math&gt;
|-
|T 
|T
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''T'''
|-
|T
|F
| style=&quot;background-color: lightblue;&quot; |
| '''F''' || T 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|-
|F
|T
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| T 
| '''T'''
|-
|F
|F
| style=&quot;background-color: lightblue;&quot; |
| '''T''' || F 
| style=&quot;background-color: pink;&quot;| F 
| '''F'''
|}

Note that this last column, column 5, is the &quot;final&quot; value for the proposition.  

----

In the presentation above, we computed entire columns all at once.  This can be nice because it is fast.

However, because this can be confusing, let's just take one row and focus on its computation.  Selecting somewhat randomly, let's focus on the second row.

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; | 
! &lt;math&gt;(\neg &lt;/math&gt;
! &lt;math&gt;P)&lt;/math&gt;
! &lt;math&gt;\land&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T 
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| F
| F
|}

To summarize how this row was computed,

1. We transcribe the values for ''P'' and ''Q'' from the left of the blue separator to the right.

2. Because of the parentheses, we first compute the negation.  It reads the T in the fourth column (under ''P'') and results in F in the third column.

3. Next we compute the conjunction, reading F on the left and T on the right.  According to the rule for conjunction, this evaluates to F.
{{robelbox|title=Exercise . Ambiguous without parens}}
Draw the truth-table for &lt;math&gt;\neg(P\land Q)&lt;/math&gt;.

Identify a final value in its truth-table, which does not match one of the final values for the truth-table of &lt;math&gt;(\neg P)\land Q&lt;/math&gt;.  

Conclude that &lt;math&gt;\neg(P\land Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\land Q&lt;/math&gt; have different semantics.  
{{robelbox/close}}
{{robelbox|title=Exercise . Conjunction associativity}}
Show that &lt;math&gt;P\land (Q\land R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land R&lt;/math&gt;.  This fact is called the &quot;associativity&quot; of conjunction.

Use this fact, without computing any more truth-tables, to argue that &lt;math&gt;P\land (Q\land (R\land S))&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\land(R\land S)&lt;/math&gt;.  
{{robelbox/close}}

The associativity of conjunction, shown in the exercise above, demonstrates that a sentence does not need parentheses when it is formed only from conjunction.  That is because we might interpret 

: &lt;math&gt;P\land Q\land R&lt;/math&gt; 

as either &lt;math&gt;(P\land Q)\land R&lt;/math&gt; or &lt;math&gt;P\land (Q\land R)&lt;/math&gt;.  But because these are equivalent, then it is not important which one we mean when we write the un-parenthesized version.  

Therefore we will write &lt;math&gt;P\land Q\land R&lt;/math&gt; without additional parentheses.  Likewise we may write &lt;math&gt;P\land Q\land R\land S&lt;/math&gt; without parentheses, because any way of parenthesizing it will always be equivalent to every other way.  

And so on, we may omit parentheses when a sentence is constructed only from conjunction, with any number of variables.  
{{robelbox|title=Exercise|theme=2}}
In SymPy one can generate the truth-table for &lt;math&gt;(\neg P)\land Q&lt;/math&gt; with the following cell.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
# Here is an implementation of the zoToTF function described in the previous 
# exercise, which we can use here to make the print-out a little nicer.
def zoToTF(lst):
    out_list = []
    for i in lst:
        if i == 0:
            out_list.append(&quot;F&quot;)
        if i == 1:
            out_list.append(&quot;T&quot;)
    return out_list

ttable = sp.logic.boolalg.truth_table( (~x)&amp;y , [x, y] )
print(&quot;x , y || (~x)&amp;y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;

Now use SymPy to print the truth-table of &lt;math&gt;\neg(P\land(\neg Q))&lt;/math&gt;.
{{robelbox/close}}

== Disjunction ==

=== Syntax ===

The syntax for disjunction is almost identical to that of conjunction.  

For &quot;''P'' or ''Q''&quot; we write &lt;math&gt;P\lor Q&lt;/math&gt;.

=== Semantics ===

The semantics of disjunction, are that the disjunction is true when one or the other disjunct is true.  This is reflected in the truth-table 

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P 
! Q
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\lor&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
| T
| T
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |
| T
| T
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |
| F
| T
| T
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |
| F
| F
| F
|}

Note that a somewhat simpler rule to remember is that &lt;math&gt;P\lor Q&lt;/math&gt; is F when both disjuncts are F.  Otherwise it is T.
{{robelbox|title=Exercise . Disjunction and negation}}
Show that &lt;math&gt;\neg(P\lor Q)&lt;/math&gt; and &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; are not equivalent.

The proof should very similar to that given for conjunction and negation, earlier.
{{robelbox/close}}
{{robelbox|title=Exercise . All disjunction}}
Decide whether parentheses are needed for a sentence constructed only from the disjunction operator.  

The answer should be decided in a process nearly identical to the one above for conjunction.
{{robelbox/close}}
{{robelbox|title= Exercise . Conjunction and disjunction}}
Investigate whether &lt;math&gt;P\land (Q\lor R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\land Q)\lor R&lt;/math&gt;.  

Use this to decide whether parentheses are needed for propositions constructed only from conjunction and disjunction.
{{robelbox/close}}
{{robelbox|title=Exercise . De Morgan's}}
The logical law of De Morgan's states that 

: &lt;math&gt;\neg(P\land Q) \equiv (\neg P)\lor (\neg Q)&lt;/math&gt;
and 
: &lt;math&gt;\neg(P\lor Q) \equiv (\neg P)\land (\neg Q)&lt;/math&gt;

This shows an important relationship between these three operators, &lt;math&gt;\neg, \land,\lor&lt;/math&gt;.  

Prove both of these claims of equivalence.  
{{robelbox/close}}
{{robelbox|title= Exercise . De Morgan's with specifics}}
Consider the sentence 
: &quot;2 is not odd and prime.&quot;

''Part 1.''

Is this sentence true?

''Part 2.''

Consider the sentence's abstraction 

: &quot;''t'' is not &lt;math&gt;O(t)&lt;/math&gt; and &lt;math&gt;P(t)&lt;/math&gt;.&quot;

Which symbolization seems most correct for the intended meaning of the original sentence?  Either 

: &lt;math&gt;(\neg O(t)) \land P(t)&lt;/math&gt;
or
: &lt;math&gt;\neg(O(t)\land P(t))&lt;/math&gt;
?

Hint: There is a subtle difference between what is communicated by the sentence &quot;2 is not odd and prime&quot; and the sentence &quot;2 is not odd and is prime&quot;.  It is precisely the difference between the two symbolizations.
{{robelbox/close}}

Here is code that prints the truth-table for disjunction in SymPy.
&lt;syntaxhighlight lang=&quot;py&quot;&gt;
ttable = sp.logic.boolalg.truth_table( x|y , [x, y] )
print(&quot;x , y || x|y&quot;)
for line in ttable:
    tfrow = zoToTF(line[0])
    print(str(tfrow) + &quot; || &quot; + str(line[1]))
&lt;/syntaxhighlight&gt;
{{robelbox|title=Exercise|theme=2}}
Use SymPy to print the truth-tables of the following.
* &lt;math&gt;\neg (P\lor Q)&lt;/math&gt;
* &lt;math&gt;(\neg P)\lor Q&lt;/math&gt;
* &lt;math&gt;(P\land Q)\lor((\neg P)\land (\neg Q))&lt;/math&gt;
{{robelbox/close}}

== Conditional ==

=== Syntax ===

The conditional symbol is &lt;math&gt;\to&lt;/math&gt;, a binary operator. Therefore the syntax for the conditional &quot;if ''P'' then ''Q''&quot; is 

: &lt;math&gt; P\to Q&lt;/math&gt;

=== Semantics ===

The semantics of the conditional are given by the truth-table

{| role=&quot;presentation&quot; class=&quot;wikitable&quot;
! P
! Q 
! style=&quot;background-color: lightblue;&quot; |
! &lt;math&gt;P&lt;/math&gt;
! &lt;math&gt;\to&lt;/math&gt;
! &lt;math&gt;Q&lt;/math&gt;
|-
|| T 
|| T 
|style=&quot;background-color: lightblue;&quot;| 
|| T
| T 
| T
|-
| T
| F
| style=&quot;background-color: lightblue;&quot; |  
| T
| F
| F
|-
| F
| T
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|-
| F
| F
| style=&quot;background-color: lightblue;&quot; |  
| F
| T
| F
|}

A simple way to describe this is: The conditional is true when the antecedent is false, or the consequent is true.

Why would this be the semantics of the conditional?  For negation, conjunction, and disjunction, the reasons for the semantics are probably close to self-evident.  But this rule for the conditional may seem not to obviously match what we think of when we think &quot;If ''P'' then ''Q''.&quot;

The important idea that we mean to capture in the semantics of the conditional, is the &quot;flow of truth&quot; from the antecedent (''P'') to the consequent (''Q'').  

Notice that, in this table, ''P'' is true on the first two rows.  Where ''Q'' is true (row 1), we say that the conditional is true.  This is because, apparently, in a sense the truth of ''P'' successfully flowed to ''Q''.  

On the other hand, where ''P'' is true and ''Q'' is false (row 2), it seems that the flow of truth from ''P'' to ''Q'' was unsuccessful.  Therefore the conditional is marked false here.

But what about the rows at which ''P'' is false (rows 3 and 4)?  Well, since ''P'' is not true then we don't require truth to &quot;flow&quot; toward ''Q'' in this scenario.  In this case, ''Q'' may be true or false, independent of ''P'', and still the conditional should be true.  

Consider the sentence &quot;If a natural number is divisible by 4 then it's divisible by 2.&quot;  This is a true sentence.  

Now consider the instance in which the number is 3.  The sentence, specialized for this instance, would be &quot;If 3 is divisible by 4 then it is divisible by 2.&quot;  This should still be a true sentence, because 3 is not divisible by 4.  In cases where the antecedent is false, we still recognize the conditional as true.  
{{robelbox|title=Exercise . Conditional equivalences}}
''Part 1.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\lor Q&lt;/math&gt; and also equivalent to &lt;math&gt;(\neg Q)\to (\neg P)&lt;/math&gt;.

''Part 2.''

Show that &lt;math&gt;P\to Q&lt;/math&gt; is ''not'' equivalent to &lt;math&gt;Q\to P&lt;/math&gt;.

''Part 3.''

Decide whether &lt;math&gt;P\to (Q\to R)&lt;/math&gt; is equivalent to &lt;math&gt;(P\to Q)\to R&lt;/math&gt;.

''Part 4.''

Decide whether &lt;math&gt;P \to Q&lt;/math&gt; is equivalent to &lt;math&gt;(\neg P)\to(\neg Q)&lt;/math&gt;.

''Part 5.''

Draw the truth-table for &lt;math&gt;(P\to Q)\land (Q\to P)&lt;/math&gt;.  

Argue that if the formula above is true then ''P'' and ''Q'' must have the same truth-value.  
{{robelbox/close}}

=== An Emphatic Caveat ===

The semantics that we give here for the conditional are, as I've also said at the beginning, the semantics of the conditional ''in mathematical settings''.

It is especially important not to confuse the semantics for this mathematical conditional, with the semantics of every kind of conditional sentence.  The conditional given by the truth-table is called the [https://en.wikipedia.org/wiki/Material_conditional| material conditional].

Some conditional sentences expression causation, like 
: &quot;If you put a magnet next to iron, then there will be an attractive force between them.&quot;

This kind of conditional sentence very much does not have the semantics given by the truth-table above.  

There are also counter-factual conditionals, like 
: &quot;If Franz Ferdinand hadn't been killed Europe wouldn't have gone to war.&quot;

Again, the semantics of such a sentence are not at all like the semantics of the truth-table.  

Later in this course we will discuss causation further, although we will not discuss counter-factuals.  To see a wide-ranging discussion of the semantics of conditionals, one could consult ''A Philosophical Guide to Conditionals'' by Bennett.

== Propositional Formulas ==

Now that we can talk about all of the symbols of propositional logic, we are in a position to define the precise definition of a propositional formula.  

The following definition is quite technical, and for some students this may be hard to process.  However, the intuitive idea of what a propositional formula is, should already be clear.  

The following are several propositional formulas.  They are just the kinds of things we made truth-tables for, earlier.

: &lt;math&gt; (P\to (\neg Q))&lt;/math&gt;
: &lt;math&gt; ((P\land Q)\land (P\lor Q))&lt;/math&gt;

Now these formulas have extra surrounding parentheses, unlike how we wrote them earlier.  These parentheses are unnecessary for us to read and understand the formula, but having them makes it easier to state a formal definition as we do below.  

But to understand how a given expression is or is not a formula, we build it up in steps.  The first step is to have a set of variables, which are the most basic formulas.

For example, consider the set of variables,

: &lt;math&gt;\mathcal F_0 = \{P,Q,R\} &lt;/math&gt;

From this we can construct the set of all formulas which are a negation of a variable.  (Soon we will rigorously define these symbols and expressions, but I want to show their use first.)

: &lt;math&gt;(\neg \mathcal F_0) = \{(\neg P),(\neg Q),(\neg R)\}&lt;/math&gt;

And the set of all conjunctions of variables.

: &lt;math&gt;\begin{aligned}
(\mathcal F_0\land \mathcal F_0) = \{&amp;(P\land P),(P\land Q),(P\land R),\\
&amp;(Q\land P),(Q\land Q),(Q\land R),\\
&amp;(R\land P),(R\land Q),(R\land R)\}
\end{aligned}&lt;/math&gt;

We may similarly compute all of the disjunctions and conditionals, written as &lt;math&gt;(\mathcal F_0\lor\mathcal F_0)&lt;/math&gt; and &lt;math&gt;(\mathcal F_0\to \mathcal F_0)&lt;/math&gt;.

If we group these all together, we get the set of all formulas which are &quot;of complexity at most 1&quot;.

: &lt;math&gt;\begin{aligned}
\mathcal F_1 = \mathcal F_0\cup &amp;(\neg \mathcal F_0) \\
&amp;(\mathcal F_0\land \mathcal F_0) \\
&amp;(\mathcal F_0\lor\mathcal F_0)\\
&amp;(\mathcal F_0\to \mathcal F_0)
\end{aligned}&lt;/math&gt;
If we compute this union, it has the elements,
: &lt;math&gt;\begin{aligned} 
\{&amp;P,Q,R,\\
&amp;(\neg P),(\neg Q),(\neg R),\\
&amp;(P\land P),(P\land Q),(P\land R),\\
&amp; \dots \\
&amp; (P\lor P),(P\lor Q),(P\lor R),\\
&amp; \dots \\
&amp; (P\to P),(P\to Q),(P\to R)\}
\end{aligned}&lt;/math&gt;

Note that this set contains 3 + 3 + 9 + 9 + 9 = 33 elements.  

We then proceed to the formulas of complexity at most 2, in a similar way.  This time we compute &lt;math&gt;(\neg \mathcal F_1)&lt;/math&gt;.  This is the set of all negations of formulas in &lt;math&gt;\mathcal F_1&lt;/math&gt;.  

Because &lt;math&gt;(\neg\mathcal F_1)&lt;/math&gt; has 33 elements, we won't try to list them.  We would also want to compute &lt;math&gt;(\mathcal F_1\land\mathcal F_1)&lt;/math&gt; which would contain &lt;math&gt;33^2 = 1089&lt;/math&gt; elements, so we want to list them even less!

But in short, the formulas of complexity at most 2 is the set 
: &lt;math&gt;\begin{aligned}
\mathcal F_2 = \mathcal F_1\cup &amp;(\neg \mathcal F_1) \\
&amp;(\mathcal F_1\land \mathcal F_1) \\
&amp;(\mathcal F_1\lor\mathcal F_1)\\
&amp;(\mathcal F_1\to \mathcal F_1)
\end{aligned}&lt;/math&gt;

This set will have a total of &lt;math&gt;33+33+33^2+33^2+33^2 = 3333&lt;/math&gt; elements.

And continuing in the same way, we may construct &lt;math&gt;\mathcal F_3,\mathcal F_4&lt;/math&gt;, and so on.

We the define the set of all propositional formulas as the union of all of these sets.

{{definition|name=propositional formula|value=
We define the set of all propositional formulas, &lt;math&gt;\mathcal F&lt;/math&gt;, in a bottom-up sequence.  This means that we progressively capture more and more formulas with each next set.

First, let &lt;math&gt;\mathcal F_0&lt;/math&gt; be any set of propositional variables.  

Next suppose that if ''X'' is any set of expressions, then by writing &lt;math&gt;\neg X&lt;/math&gt; we mean 
: &lt;math&gt;(\neg X) = \{(\neg Q) : Q\in X\}&lt;/math&gt;
Likewise define, for any sets of expressions ''X'' and ''Y'', 
: &lt;math&gt;\begin{aligned}
(X\land Y) &amp;= \{(P\land Q):P\in X, Q\in Y\}\\
(X\lor Y) &amp;= \{(P\lor Q): P\in X, Q\in Y\}\\
(X\to Y) &amp;= \{(P\to Q): P\in X, Q\in Y\}
\end{aligned}&lt;/math&gt;

Now if &lt;math&gt;\mathcal F_n&lt;/math&gt; is defined for any natural number ''n'', then we use this to define &lt;math&gt;\mathcal F_{n+1}&lt;/math&gt; as follows.
: &lt;math&gt;\begin{aligned}
\mathcal F_{n+1} = \mathcal F_n&amp;\cup (\neg \mathcal F_n)\\
&amp;\cup (\mathcal F_n\land \mathcal F_n) \\
&amp;\cup (\mathcal F_n\lor \mathcal F_n)\\
&amp;\cup (\mathcal F_n\to\mathcal F_n)
\end{aligned}&lt;/math&gt;

Finally, define &lt;math&gt;\mathcal F = \bigcup_{n=0}^\infty F_n&lt;/math&gt; which is the '''set of all propositional formulas from &lt;math&gt;\mathcal F_0&lt;/math&gt;'''.  

Then any expression, &lt;math&gt;\varphi&lt;/math&gt;, is a '''propositional formula''' if &lt;math&gt;\varphi\in\mathcal F&lt;/math&gt;.
}}
{{robelbox|title=Exercise . Compute a few levels|theme=2}}
Set &lt;math&gt;\mathcal F_0=\{P, Q\}&lt;/math&gt;.  

Compute &lt;math&gt;\neg \mathcal F_0&lt;/math&gt; and &lt;math&gt;\mathcal F_1&lt;/math&gt;. 

Then repeat the exercise if &lt;math&gt;\mathcal F_0 = \{P\}&lt;/math&gt;.  
{{robelbox/close}}
This bottom-up construction of all formulas makes it easy to demonstrate that an expression is a formula.  For example, we can show that &lt;math&gt;((\neg P)\lor Q)&lt;/math&gt; by the following argument.

First, &lt;math&gt;P\in \mathcal F_0&lt;/math&gt; and therefore &lt;math&gt;(\neg P)\in\mathcal F_1&lt;/math&gt;.  

Second, &lt;math&gt;Q\in\mathcal F_0&lt;/math&gt; and therefore &lt;math&gt;Q\in \mathcal F_1&lt;/math&gt;.

Finally, because &lt;math&gt;(\neg P)\in \mathcal F_1&lt;/math&gt; and &lt;math&gt;Q\in\mathcal F_1&lt;/math&gt;, therefore &lt;math&gt;((\neg P)\lor Q)\in\mathcal F_2&lt;/math&gt;.  

This last shows that &lt;math&gt;((\neg P)\lor Q)&lt;/math&gt; is a formula, because it occurs in one of the sets &lt;math&gt;\mathcal F_n&lt;/math&gt; for some &lt;math&gt;n\le 0&lt;/math&gt;.
{{robelbox|title=Exercise|theme=2}}
Show that &lt;math&gt;((\neg P)\land(Q\to R)&lt;/math&gt; is a formula.

Also show that &lt;math&gt;\neg P&lt;/math&gt; is ''not'' a formula.
{{robelbox/close}}

Technically &lt;math&gt;\neg P&lt;/math&gt; is not a formula.  In order to be a formula it must have the final surrounding parentheses, &lt;math&gt;(\neg P)&lt;/math&gt;.  

However, these final surrounding parentheses are only useful in having a simple definition of a formula, and are not actually important for humans to read the formula.  

Therefore, we will write formulas without the final surrounding parentheses.  We will just understand that, technically, the final parentheses need to be there, but for simplicity we just omit them.</text>
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    <title>User:Guy vandegrift/sandbox/Archives/2</title>
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    <revision>
      <id>2624863</id>
      <timestamp>2024-05-02T23:43:18Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>New resource with &quot;==Pierogi dough==  ==Surreal numbers==&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38" xml:space="preserve">==Pierogi dough==

==Surreal numbers==</text>
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    <revision>
      <id>2624864</id>
      <parentid>2624863</parentid>
      <timestamp>2024-05-02T23:43:46Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
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      <text bytes="49" xml:space="preserve">{{Header}}
==Pierogi dough==

==Surreal numbers==</text>
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    <revision>
      <id>2624866</id>
      <parentid>2624864</parentid>
      <timestamp>2024-05-02T23:44:21Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <minor />
      <comment>Guy vandegrift moved page [[Toggle the table of contents User:Guy vandegrift/sandbox/Archives/2]] to [[User:Guy vandegrift/sandbox/Archives/2]] without leaving a redirect</comment>
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==Pierogi dough==

==Surreal numbers==</text>
      <sha1>ky526gnow7s7r1qoroal0ltpotqt6lm</sha1>
    </revision>
    <revision>
      <id>2624873</id>
      <parentid>2624866</parentid>
      <timestamp>2024-05-02T23:51:38Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Surreal numbers */</comment>
      <model>wikitext</model>
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      <text bytes="696" xml:space="preserve">{{Header}}
==Pierogi dough==

==Surreal numbers table==
Rescued from first attempt (and discarded by me)
{| class=&quot;wikitable floatleft&quot;
|+
!Day
!#
!Set
|-
|0
|0
|&lt;nowiki&gt;{|}&lt;/nowiki&gt;
|-
| rowspan=&quot;2&quot; |1
| -1
|&lt;nowiki&gt;{|0}&lt;/nowiki&gt;
|-
|1
|&lt;nowiki&gt;{0|}&lt;/nowiki&gt;
|-
| rowspan=&quot;4&quot; |2
| -2
|&lt;nowiki&gt;{|-1}&lt;/nowiki&gt;
|-
| -1/2
|&lt;nowiki&gt;{-1|0}&lt;/nowiki&gt;
|-
|1/2
|&lt;nowiki&gt;{0|1}&lt;/nowiki&gt;
|-
|2
|&lt;nowiki&gt;{1|}&lt;/nowiki&gt;
|-
| rowspan=&quot;8&quot; |3
| -3
|&lt;nowiki&gt;{|-2}&lt;/nowiki&gt;
|-
| -3/2
|&lt;nowiki&gt;{-2|-1}&lt;/nowiki&gt;
|-
| -3/4
|&lt;nowiki&gt;{-1|-1/2}&lt;/nowiki&gt;
|-
| -1/4
|&lt;nowiki&gt;{-1/2|0}&lt;/nowiki&gt;
|-
|1/4
|&lt;nowiki&gt;{0|1/2}&lt;/nowiki&gt;
|-
|3/4
|&lt;nowiki&gt;{1/2|1}&lt;/nowiki&gt;
|-
|3/2
|&lt;nowiki&gt;{1|2}&lt;/nowiki&gt;
|-
|3
|&lt;nowiki&gt;{2|}&lt;/nowiki&gt;
|}</text>
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    <revision>
      <id>2624875</id>
      <parentid>2624873</parentid>
      <timestamp>2024-05-02T23:53:32Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Pierogi dough */</comment>
      <model>wikitext</model>
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      <text bytes="2445" xml:space="preserve">{{Header}}
==Pierogi dough==
If you are looking for Draftify improvements, they are already submitted as {{tl|draftify}}

Here's a recipe for pierogi dough that uses just flour, salt, egg, butter, and milk:

'''Ingredients:'''

* 3 1/2 cups all-purpose flour
* 1/2 teaspoon salt
* 1 cup milk, scalded (not boiling)
* 1/2 stick (4 tablespoons) butter
* 2 large eggs, beaten

'''Instructions:'''

# In a large bowl, whisk together the flour and salt.
# In a separate saucepan, heat the milk until hot but not boiling (scalded). Add the butter to the hot milk and let it melt.
# Pour the hot milk mixture into the flour mixture and stir well with a wooden spoon until a dough forms.

* You can also use a stand mixer with a dough hook for this step.

# Add the beaten eggs and continue mixing until the dough comes together and is relatively smooth. It will still be a bit sticky.
# Turn the dough out onto a lightly floured surface and knead for 5-10 minutes, or until the dough is smooth and elastic. Add a bit more flour, 1 tablespoon at a time, if the dough is too sticky.
# Form the dough into a ball, cover it with a damp cloth or plastic wrap, and let it rest at room temperature for 20 minutes.
# After resting, you can roll out the dough and make your pierogi!

'''Tips:'''

* Don't overwork the dough, as this can make it tough.
* The dough can be chilled for up to 2 hours before using.
* If you don't have buttermilk, you can make your own by adding 1 tablespoon of vinegar or lemon juice to 1 cup of milk and letting it sit for 5 minutes.

This recipe is from Moja miłość do Polski: [invalid URL removed], a website dedicated to Polish food and culture. They claim this is a tried-and-tested family recipe that yields delicious and easy-to-work-with pierogi dough.

==Surreal numbers table==
Rescued from first attempt (and discarded by me)
{| class=&quot;wikitable floatleft&quot;
|+
!Day
!#
!Set
|-
|0
|0
|&lt;nowiki&gt;{|}&lt;/nowiki&gt;
|-
| rowspan=&quot;2&quot; |1
| -1
|&lt;nowiki&gt;{|0}&lt;/nowiki&gt;
|-
|1
|&lt;nowiki&gt;{0|}&lt;/nowiki&gt;
|-
| rowspan=&quot;4&quot; |2
| -2
|&lt;nowiki&gt;{|-1}&lt;/nowiki&gt;
|-
| -1/2
|&lt;nowiki&gt;{-1|0}&lt;/nowiki&gt;
|-
|1/2
|&lt;nowiki&gt;{0|1}&lt;/nowiki&gt;
|-
|2
|&lt;nowiki&gt;{1|}&lt;/nowiki&gt;
|-
| rowspan=&quot;8&quot; |3
| -3
|&lt;nowiki&gt;{|-2}&lt;/nowiki&gt;
|-
| -3/2
|&lt;nowiki&gt;{-2|-1}&lt;/nowiki&gt;
|-
| -3/4
|&lt;nowiki&gt;{-1|-1/2}&lt;/nowiki&gt;
|-
| -1/4
|&lt;nowiki&gt;{-1/2|0}&lt;/nowiki&gt;
|-
|1/4
|&lt;nowiki&gt;{0|1/2}&lt;/nowiki&gt;
|-
|3/4
|&lt;nowiki&gt;{1/2|1}&lt;/nowiki&gt;
|-
|3/2
|&lt;nowiki&gt;{1|2}&lt;/nowiki&gt;
|-
|3
|&lt;nowiki&gt;{2|}&lt;/nowiki&gt;
|}</text>
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  <page>
    <title>File:0502quadratic02.png</title>
    <ns>6</ns>
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    <revision>
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      <timestamp>2024-05-02T23:45:25Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>{{Information
|Description=Graph of quadratic function showing where slope is vertical and where slope is 0.5.
|Source={{own}}
|Date=2024-05-02
|Author=ThaniosAkro
|Permission=public domain
}}</comment>
      <model>wikitext</model>
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      <text bytes="234" xml:space="preserve">== Summary ==
{{Information
|Description=Graph of quadratic function showing where slope is vertical and where slope is 0.5.
|Source={{own}}
|Date=2024-05-02
|Author=ThaniosAkro
|Permission=public domain
}}
== Licensing ==
{{PD-self}}</text>
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  <page>
    <title>User talk:Wpu23</title>
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      <timestamp>2024-05-03T03:33:38Z</timestamp>
      <contributor>
        <username>Wpu23</username>
        <id>2984437</id>
      </contributor>
      <comment>creating user page</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="0" />
      <sha1>phoiac9h4m842xq45sp7s6u21eteeq1</sha1>
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  <page>
    <title>User talk:2601:18D:4800:EA0:B804:B790:ACBB:33F4</title>
    <ns>3</ns>
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    <revision>
      <id>2624947</id>
      <timestamp>2024-05-03T07:42:24Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>New resource with &quot;== May 2024 == {{subst:uw-vandalism1}} ~~~~&quot;</comment>
      <model>wikitext</model>
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      <text bytes="499" xml:space="preserve">== May 2024 ==
[[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Help desk|help desk]]. Thank you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:42, 3 May 2024 (UTC)</text>
      <sha1>l4hj1rgsn8xvdnlbmoornx7t2js377m</sha1>
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  <page>
    <title>User talk:204.219.240.33</title>
    <ns>3</ns>
    <id>305019</id>
    <revision>
      <id>2624949</id>
      <timestamp>2024-05-03T07:42:58Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>New resource with &quot;== May 2024 ==  {{subst:uw-vandalism1}} ~~~~&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="500" xml:space="preserve">== May 2024 == 
[[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Help desk|help desk]]. Thank you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:42, 3 May 2024 (UTC)</text>
      <sha1>o4mc1fpl9nwwvrxnsdkptqe97vnlqnj</sha1>
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  <page>
    <title>User talk:2406:7400:FF03:FED7:2498:128:A430:503F</title>
    <ns>3</ns>
    <id>305020</id>
    <revision>
      <id>2624950</id>
      <timestamp>2024-05-03T07:43:19Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>New resource with &quot;== May 2024 ==  {{subst:uw-vandalism1}} ~~~~&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="500" xml:space="preserve">== May 2024 == 
[[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Help desk|help desk]]. Thank you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:43, 3 May 2024 (UTC)</text>
      <sha1>bjr8x9xdxynvgj1z4yew0z9twk11c6t</sha1>
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  </page>
  <page>
    <title>Category:Toilets</title>
    <ns>14</ns>
    <id>305021</id>
    <revision>
      <id>2624952</id>
      <timestamp>2024-05-03T08:07:04Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>New resource with &quot;[[Category:Hygiene]] [[Category:Public health]]&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47" xml:space="preserve">[[Category:Hygiene]]
[[Category:Public health]]</text>
      <sha1>4n4ixhs0fid9uwyawgz4z70907r5a6q</sha1>
    </revision>
  </page>
  <page>
    <title>Schlangenköpfe und -körper</title>
    <ns>0</ns>
    <id>305022</id>
    <revision>
      <id>2624964</id>
      <timestamp>2024-05-03T11:34:13Z</timestamp>
      <contributor>
        <username>Davud Kilic</username>
        <id>2983088</id>
      </contributor>
      <comment>New resource with &quot;[[File:Presenation Davud Kilic.pdf|thumb|This is my Presentation about my dataset]]&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="83" xml:space="preserve">[[File:Presenation Davud Kilic.pdf|thumb|This is my Presentation about my dataset]]</text>
      <sha1>hnmuuus1xsqiihkczo6s2g27spyfrjd</sha1>
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